<PAGE>   1
                                                                    EXRHIBIT 4.1


                          REGISTRATION RIGHTS AGREEMENT

         This Registration Rights Agreement is dated as of June 6, 2001 by and
between Torch Offshore, Inc., a Delaware corporation (together with its
successors and assigns, the "Company"), Friends of Lime Rock LP, a Delaware
limited partnership ("Friends of Lime Rock"), and Riverside Investments LLC, a
Delaware limited liability company ("Riverside").

         1. Background. The Company has agreed to issue to Friends of Lime Rock
and Riverside certain shares of the Company's common stock, par value $.01 per
share (the "Common Stock"), pursuant to the transactions contemplated by the
Contribution Agreement dated as of January 15, 2001 among the Company, Friends
of Lime Rock, Riverside and Torch, Inc. (the "Contribution Agreement").

         2. Registration under Securities Act, etc.

         2.1. Registration on Request.

                  (a) Concurrently with or from time to time after the date
         hereof, upon the written request of any Holder of a majority of the
         Registrable Securities (a "Requesting Holder"), requesting that the
         Company effect the registration under the Securities Act of all or a
         portion of such Requesting Holder's Registrable Securities and
         specifying the intended method of disposition thereof and whether or
         not such requested registration is to be an underwritten offering,
         subject to the limitations set forth in subsection (e) of this Section
         2.1, the Company will use its best efforts to effect the registration
         under the Securities Act of the Registrable Securities which the
         Company has been so requested to register by the Requesting Holder and
         any other Holder joining in such request (which request shall specify
         the intended method of disposition of such Registrable Securities), to
         the extent requisite to permit the disposition (in accordance with the
         intended methods thereof as aforesaid) of the Registrable Securities so
         to be registered (a "Demand Registration").

                  (b) Registration of Other Securities. Whenever the Company
         shall effect a Demand Registration pursuant to this Section 2.1 in
         connection with an underwritten offering, no securities other than
         Registrable Securities shall be included among the securities covered
         by such registration unless (i) the managing underwriter of such
         offering shall have advised the Requesting Holder in writing that the
         inclusion of such other securities would not adversely affect such
         offering or (ii) the Requesting Holder shall have consented in writing
         to the inclusion of such other securities.

                  (c) Registration Statement Form. A Demand Registration under
         this Section 2.1 shall be on such appropriate registration form of the
         Commission (i) as shall be selected by the Company and as shall be
         reasonably acceptable to the Requesting Holder, and (ii) as shall
         permit the disposition of such Registrable Securities in accordance
         with the intended method or methods of disposition specified in the
         request for such

<PAGE>   2

         registration. The Company agrees to include in any such registration
         statement all information which the Requesting Holder shall reasonably
         request.

                  (d) Expenses. The Company will pay all Registration Expenses
         in connection with any Demand Registration requested pursuant to this
         Section 2.1. Selling Expenses in connection with this Section 2.1 shall
         be incurred by the participating Holders.

                  (e) Limitations on Requested Registrations. The Company's
         obligation to take or continue any action to effect a Demand
         Registration under this Section 2.1 shall be subject to the following:

                           (i) The Company shall not be required to effect more
                  than three Demand Registrations pursuant to this Section 2.1.

                           (ii) The Company shall not be required to effect a
                  Demand Registration pursuant to this Section 2.1 unless the
                  Requesting Holder (together with any other Holder joining in
                  such request) has requested the registration of a number of
                  Registrable Securities the gross sale proceeds of which are
                  reasonably expected to be at least $5 million.

                           (iii) The Company shall not be required to effect a
                  Demand Registration pursuant to this Section 2.1 during the
                  period of any lock-up agreement entered into pursuant to
                  Section 2.4(c).

                           (iv) The Company shall not be required to effect a
                  Demand Registration pursuant to this Section 2.1 after the
                  expiration of 10 years from the date hereof.

                           (v) The Company shall not be obligated to effect a
                  Demand Registration within six months after the effective date
                  of a previous Demand Registration.

                           (vi) The Company shall not be obligated to effect a
                  Demand Registration if the Company shall furnish to the
                  Requesting Holder a certificate signed by the Company's
                  Chairman of the Board stating that in the good faith judgment
                  of the Board of Directors of the Company, it would be
                  materially harmful to the economic prospects of the Company
                  for such Demand Registration to be effected at such time, in
                  which event the Company shall have the right to defer such
                  filing for a period of not more than 45 days after receipt of
                  the initial request for the Demand Registration.

                  (f) Selection of Underwriters. If a Demand Registration
         pursuant to this Section 2.1 involves an underwritten offering, the
         underwriter or underwriters thereof shall be selected by the Company,
         with the reasonable approval of the Requesting Holder.

                                      -2-
<PAGE>   3

         2.2. Incidental Registration.

                  (a) Right to Include Registrable Securities. If the Company at
         any time proposes to register any of its securities under the
         Securities Act (other than (i) in connection with a registration
         statement on Form S-8 or Form S-4 or any similar successor form and the
         registration form to be used may be used for the registration of
         Registrable Securities, or (ii) pursuant to Section 2.1) (an
         "Incidental Registration"), whether or not for sale for its own
         account, it will each such time give prompt written notice to the
         Holders of its intention to do so and of the Holders' rights under this
         Section 2.2. Upon the written request of the any Holder made within ten
         (10) days after the receipt of any such notice (which request shall
         specify the Registrable Securities intended to be disposed of and the
         intended method of disposition thereof), the Company will use its best
         efforts to effect the registration under the Securities Act of all
         Registrable Securities that the Company has been so requested to
         register, to the extent requisite to permit the disposition (in
         accordance with the intended methods thereof as aforesaid) of the
         Registrable Securities so to be registered, provided that if, at any
         time after giving written notice of its intention to register any
         securities and prior to the effective date of the registration
         statement filed in connection with such registration, the Company shall
         determine for any reason not to register or to delay registration of
         such securities, the Company may, at its election, give written notice
         of such determination to the Holders participating in such registration
         and, thereupon, (i) in the case of a determination not to register,
         shall be relieved of its obligation to register any Registrable
         Securities in connection with such registration (but not from its
         obligation to pay the Registration Expenses in connection therewith),
         without prejudice, however, to the rights of the Holders to request
         that such registration be effected as a Demand Registration under
         Section 2.1, and (ii) in the case of a determination to delay
         registering, shall be permitted to delay registering any Registrable
         Securities, for the same period as the delay in registering such other
         securities. No registration effected under this Section 2.2 shall be
         deemed to have been effected pursuant to Section 2.1 or shall relieve
         the Company of its obligation to effect any registration upon request
         under Section 2.1. The Company will pay all Registration Expenses in
         connection with each registration of Registrable Securities requested
         pursuant to this Section 2.2 and any Selling Expenses shall be incurred
         by the participating Holders.

                  (b) Priority in Incidental Registrations.

                           (i) If (A) a registration pursuant to this Section
                  2.2 involves an underwritten offering of the securities so
                  being registered for sale for the account of a stockholder
                  (other than the Holders) exercising a demand registration
                  right pursuant to another registration rights agreement to be
                  distributed by or through one or more underwriters of
                  recognized standing under underwriting terms appropriate for
                  such a transaction, and (B) the managing underwriter of such
                  underwritten offering informs the Company and the Holders in
                  writing of its opinion that the number of securities requested
                  to be included in such registration will adversely affect the
                  success of such offering, then the Company will include in
                  such registration, to the extent of the number that the
                  Company is so advised can be sold in (or during the time of)
                  such offering: first, such securities proposed

                                      -3-

<PAGE>   4

                  by the stockholder exercising the demand registration right to
                  be sold for its account; second, such securities requested to
                  be included pursuant to incidental registration rights in such
                  registration by the holder or holders, as the case may be,
                  including the Holders, pro rata on the basis of the number of
                  such securities so proposed to be sold by all such security
                  holders and so requested to be included; and third, such
                  securities proposed by the Company to be sold for its own
                  account.


                           (ii) If (A) a registration pursuant to this Section
                  2.2 involves an underwritten offering of the securities so
                  being registered for sale for the account of the Company, to
                  be distributed by or through one or more underwriters of
                  recognized standing under underwriting terms appropriate for
                  such a transaction, and (B) the managing underwriter of such
                  underwritten offering shall inform the Company and the Holders
                  in writing of its opinion that the number of securities
                  requested to be included in such registration will adversely
                  affect the success of such offering, then the Company will
                  include in such registration, to the extent of the number that
                  the Company is so advised can be sold in (or during the time
                  of) such offering: first, securities proposed by the Company
                  to be sold for its own account; and second, such securities
                  requested to be included pursuant to incidental registration
                  rights in such registration by the holder or holders, as the
                  case may be, including the Holders, pro rata on the basis of
                  the number of such securities so proposed to be sold by all
                  such security holders and so requested to be included.

         2.3. Registration Procedures. If and whenever the Company is required
to use its best efforts to effect the registration of any Registrable Securities
under the Securities Act as provided in Sections 2.1 and 2.2, the Company will
as expeditiously as possible:

                  (a) prepare and file with the Commission the requisite
         registration statement to effect such registration and thereafter use
         its best efforts to cause such registration statement to become
         effective, provided that the Company may discontinue any registration
         of its securities that are not Registrable Securities (and, under the
         circumstances specified in Section 2.2(a), its securities that are
         Registrable Securities) at any time prior to the effective date of the
         registration statement relating thereto;

                  (b) prepare and file with the Commission such amendments and
         supplements to such registration statement and the prospectus used in
         connection therewith as may be necessary to keep such registration
         statement effective and to comply with the provisions of the Securities
         Act with respect to the disposition of all securities covered by such
         registration statement until the earlier of (i) 180 days following the
         effectiveness of such registration statement or (ii) such time as all
         of such securities have been disposed of in accordance with the
         intended methods of disposition by the seller or sellers thereof set
         forth in such registration statement;

                  (c) furnish to the Holders such number of conformed copies of
         such registration statement and of each such amendment and supplement
         thereto (in each case including all exhibits), such number of copies of
         the prospectus contained in such

                                      -4-
<PAGE>   5

         registration statement (including each preliminary prospectus and any
         summary prospectus) and any other prospectus filed under Rule 424 under
         the Securities Act, in conformity with the requirements of the
         Securities Act, and such other documents, as the Holders may reasonably
         request;

                  (d) use its best efforts to register or qualify all
         Registrable Securities and other securities covered by such
         registration statement under such other securities or blue sky laws of
         such jurisdictions as the Holders shall reasonably request, to keep
         such registration or qualification in effect for so long as such
         registration statement remains in effect, and take any other action
         that may be reasonably necessary or advisable to enable the Holders to
         consummate the disposition in such jurisdictions of the securities
         owned by it, except that the Company shall not for any such purpose be
         required to qualify generally to do business as a foreign corporation
         in any jurisdiction wherein it would not but for the requirements of
         this subdivision (d) be obligated to be so qualified, to subject itself
         to taxation in any such jurisdiction or to consent to general service
         of process in any such jurisdiction;

                  (e) use its best efforts to cause all Registrable Securities
         covered by such registration statement to be registered with or
         approved by such other governmental agencies or authorities as may be
         necessary to enable the seller or sellers thereof to consummate the
         disposition of such Registrable Securities;

                  (f) in an underwritten offering, furnish to the Holders a
         signed counterpart, addressed to such seller (and underwriters, if any)
         of:

                           (i) an opinion of counsel for the Company, dated the
                  effective date of such registration statement (and, if such
                  registration includes an underwritten public offering, dated
                  the date of the closing under the underwriting agreement),
                  reasonably satisfactory in form and substance to such seller,
                  and

                           (ii) a "comfort" letter, dated the effective date of
                  such registration statement (and, if such registration
                  includes an underwritten public offering, dated the date of
                  the closing under the underwriting agreement), signed by the
                  independent public accountants who have certified the
                  Company's financial statements included in such registration
                  statement,

         covering substantially the same matters with respect to such
         registration statement (and the prospectus included therein) and, in
         the case of the accountants' letter, with respect to events subsequent
         to the date of such financial statements, as are customarily covered in
         opinions of issuer's counsel and in accountants' letters delivered to
         the underwriters in underwritten public offerings of securities and, in
         the case of the accountants' letter, such other financial matters, and,
         in the case of the legal opinion, such other legal matters, as such
         seller may reasonably request;

                  (g) notify the Holders, at any time when a prospectus relating
         thereto is required to be delivered under the Securities Act, upon
         discovery that, or upon the happening of any event as a result of
         which, the prospectus included in such registration

                                      -5-
<PAGE>   6

         statement, as then in effect, includes an untrue statement of a
         material fact or omits to state any material fact required to be stated
         therein or necessary to make the statements therein not misleading in
         the light of the circumstances under which they were made, and at the
         request of the Holders promptly prepare and furnish to the Holders a
         reasonable number of copies of a supplement to or an amendment of such
         prospectus as may be necessary so that, as thereafter delivered to the
         purchasers of such securities, such prospectus shall not include an
         untrue statement of a material fact or omit to state a material fact
         required to be stated therein or necessary to make the statements
         therein not misleading in the light of the circumstances under which
         they were made;

                  (h) otherwise use its best efforts to comply with all
         applicable rules and regulations of the Commission, and make available
         to its security holders, as soon as reasonably practicable, an earnings
         statement covering the period of at least twelve months, but not more
         than eighteen months, beginning with the first full calendar month
         after the effective date of such registration statement, which earnings
         statement shall satisfy the provisions of Section 11(a) of the
         Securities Act;

                  (i) furnish to the Holders at least five (5) business days
         prior to the filing thereof a copy of any amendment or supplement to
         such registration statement or prospectus and shall not file any
         thereof to which the Holders shall have reasonably objected on the
         grounds that such amendment or supplement does not comply in all
         material respects with the requirements of the Securities Act or of the
         rules or regulations thereunder;

                  (j) provide and cause to be maintained a transfer agent and
         registrar for all Registrable Securities covered by such registration
         statement from and after a date not later than the effective date of
         such registration statement;

                  (k) use its best efforts to list all Registrable Securities
         covered by such registration statement on any securities exchange or
         automated quotation system on which any of the Registrable Securities
         is then listed; and

                  (l) (enter into such agreements and take such other actions as
         the Holders shall reasonably request in order to expedite or facilitate
         the disposition of such Registrable Securities.

The Company may require the Holders participating in a registration to furnish
the Company such information regarding such Holder and the distribution of such
securities as the Company may from time to time reasonably request in writing.

         Each Holder agrees that upon receipt of any notice from the Company of
the happening of any event of the kind described in subdivision (g) of this
Section 2.3, the Holder will forthwith discontinue its disposition of
Registrable Securities pursuant to the registration statement relating to such
Registrable Securities until the Holder's receipt of the copies of the
supplemented or amended prospectus contemplated by subdivision (g) of this
Section 2.3 and, if so directed by the Company, will deliver to the Company (at
the Company's expense) all copies, other than

                                      -6-
<PAGE>   7

permanent file copies, then in the Holder's possession of the prospectus
relating to such Registrable Securities current at the time of receipt of such
notice.

         2.4. Underwritten Offerings; Lockup Agreements.

                  (a) Requested Underwritten Offerings. If requested by the
         underwriters for any underwritten offering by a Holder pursuant to a
         Demand Registration requested under Section 2.1, the Company will enter
         into an underwriting agreement with such underwriters for such
         offering, such agreement to be reasonably satisfactory in substance and
         form to the Company, the Holder and the underwriters and to contain
         such representations and warranties by the Company and such other terms
         as are generally prevailing in agreements of this type, including,
         without limitation, indemnities to the effect and to the extent
         provided in Section 2.6 and customary contribution provisions, and a
         customary "lock-up" agreement not to sell or otherwise dispose of
         Common Stock or securities exercisable, exchangeable or convertible
         therefore or derivatives thereof for up to 180 days from the date of
         any prospectus used in such an offering, on the terms and to the extent
         reasonably requested by the underwriters with respect to such offering.
         The Holder shall be a party to such underwriting agreement and agrees
         to (a) sell its securities on the basis provided in any underwriting
         agreements and (b) complete and execute all customary questionnaires,
         underwriting agreements and other documents reasonably required under
         the terms of such underwriting agreements. The Holders shall not be
         required to make any representations or warranties to the Company or
         the underwriters other than representations, warranties regarding the
         Holders, its Registrable Securities and its intended method of
         distribution and any other representation required by law.

                  (b) Incidental Underwritten Offerings. If the Company at any
         time proposes to register any of its securities under the Securities
         Act whether or not for sale for its own account as contemplated by
         Section 2.2 and such securities are to be distributed by or through one
         or more underwriters, the Company will, if requested by the Holders as
         provided in Section 2.2 and subject to the provisions of Section
         2.2(b), arrange for such underwriters to include all the Registrable
         Securities to be offered and sold by the Holders among the securities
         to be distributed by such underwriters. The participating Holders shall
         be a party to the underwriting agreement between the Company and such
         underwriters and agrees to (a) sell its securities on the basis
         provided in any underwriting agreements and (b) complete and execute
         all customary questionnaires, underwriting agreements and other
         documents reasonably required under the terms of such underwriting
         agreements. The Holders shall not be required to make any
         representations or warranties to the Company or the underwriters other
         than representations, warranties regarding the Holders, its Registrable
         Securities and its intended method of distribution and any other
         representation required by law.

                  (c) Lockup by Holders. Each holder will, in connection with
         the Company's initial public offering of its Common Stock, agree to a
         customary lock-up agreement with respect to sales by it of Registrable
         Securities for up to 180 days following the pricing of such an
         offering, if and to the same extent as shall be agreed to by at least a
         majority of the other stockholders of the Company.

                                      -7-
<PAGE>   8

         2.5. Preparation; Reasonable Investigation. In connection with the
preparation and filing of each registration statement under the Securities Act
pursuant to this Agreement, the Company will give the Holders, and their counsel
and accountants, the opportunity to participate in the preparation of such
registration statement, each prospectus included therein or filed with the
Commission, and each amendment thereof or supplement thereto, and will give it
such access to its books and records and such opportunities to discuss the
business of the Company with its officers and the independent public accountants
who have certified its financial statements as shall be necessary, in the
opinion of the Holders' counsel, to conduct a reasonable investigation within
the meaning of the Securities Act.

         2.6. Indemnification.

                  (a) Indemnification by the Company. In the event of any
         registration of any securities of the Company under the Securities Act,
         the Company will, and hereby does, in the case of any registration
         statement filed pursuant to Section 2.1 or 2.2, indemnify and hold
         harmless the Holders, its directors and officers, and each other
         Person, if any, who controls any Holder within the meaning of the
         Securities Act against any losses, claims, damages or liabilities,
         joint or several, to which the Holders or any such director or officer
         or controlling person may become subject under the Securities Act or
         otherwise, insofar as such losses, claims, damages or liabilities (or
         actions or proceedings, whether commenced or threatened, in respect
         thereof) arise out of or are based upon any untrue statement or alleged
         untrue statement of any material fact contained in any registration
         statement under which such securities were registered under the
         Securities Act, any preliminary prospectus, final prospectus or summary
         prospectus contained therein, or any amendment or supplement thereto,
         or any omission or alleged omission to state therein a material fact
         required to be stated therein or necessary to make the statements
         therein not misleading, and the Company will reimburse the Holders and
         each such director, officer, and controlling person for any legal or
         any other expenses reasonably incurred by them in connection with
         investigating or defending any such loss, claim, liability, action or
         proceeding; provided that the Company shall not be liable in any such
         case to the extent that any such loss, claim, damage, liability (or
         action or proceeding in respect thereof) or expense arises out of or is
         based upon an untrue statement or alleged untrue statement or omission
         or alleged omission made in such registration statement, any such
         preliminary prospectus, final prospectus, summary prospectus, amendment
         or supplement in reliance upon and in conformity with written
         information furnished to the Company by a Holder specifically stating
         that it is for use in the preparation thereof. Such indemnity shall
         remain in full force and effect regardless of any investigation made by
         or on behalf of any Holder or any such director, officer, underwriter
         or controlling person and shall survive the transfer of such securities
         by such Holder.

                  (b) Indemnification by the Holders. The Company may require,
         as a condition to including any Registrable Securities in any
         registration statement filed pursuant to Section 2.6, that the Company
         shall have received an undertaking satisfactory to it from each
         participating Holder, to indemnify and hold harmless (in the same
         manner and to the same extent as set forth in subdivision (a) of this
         Section 2.6) the Company, each director of the Company, each officer of
         the Company and each other Person, if any,

                                      -8-
<PAGE>   9

         who controls the Company within the meaning of the Securities Act, with
         respect to any statement or alleged statement in or omission or alleged
         omission from such registration statement, any preliminary prospectus,
         final prospectus or summary prospectus contained therein, or any
         amendment or supplement thereto, if such statement or alleged statement
         or omission or alleged omission was made in reliance upon and in
         conformity with written information furnished to the Company by the
         Holder specifically stating that it is for use in the preparation of
         such registration statement, preliminary prospectus, final prospectus,
         summary prospectus, amendment or supplement. Such indemnity shall
         remain in full force and effect, regardless of any investigation made
         by or on behalf of the Company or any such director, officer or
         controlling Person and shall survive the transfer of such securities by
         any Holder.

                  (c) Notices of Claims, etc. Promptly after receipt by an
         indemnified party of notice of the commencement of any action or
         proceeding involving a claim referred to in the preceding subdivisions
         of this Section 2.6, such indemnified party will, if a claim in respect
         thereof is to be made against an indemnifying party, give written
         notice to the latter of the commencement of such action, provided that
         the failure of any indemnified party to give notice as provided herein
         shall not relieve the indemnifying party of its obligations under the
         preceding subdivisions of this Section 2.6, except to the extent that
         the indemnifying party is actually prejudiced by such failure to give
         notice. In case any such action is brought against an indemnified
         party, unless in such indemnified party's reasonable judgment a
         conflict of interest between such indemnified and indemnifying parties
         may exist in respect of such claim, the indemnifying party shall be
         entitled to participate in and to assume the defense thereof, jointly
         with any other indemnifying party similarly notified to the extent that
         it may wish, with counsel reasonably satisfactory to such indemnified
         party, and after notice from the indemnifying party to such indemnified
         party of its election so to assume the defense thereof, the
         indemnifying party shall not be liable to such indemnified party for
         any legal or other expenses subsequently incurred by the latter in
         connection with the defense thereof other than reasonable costs of
         investigation. No indemnifying party shall, without the consent of the
         indemnified party, consent to entry of any judgment or enter into any
         settlement that does not include as an unconditional term thereof the
         giving by the claimant or plaintiff to such indemnified party of a
         release from all liability in respect to such claim or litigation.

                  (d) Indemnification Payments. The indemnification required by
         this Section 2.6 shall be made by periodic payments of the amount
         thereof during the course of the investigation or defense, as and when
         bills are received or expense, loss, damage or liability is incurred.

         3. Definitions. As used herein, unless the context otherwise requires,
the following terms have the following respective meanings:

                  COMMISSION: The Securities and Exchange Commission or any
         other federal agency at the time administering the Securities Act.

                                      -9-
<PAGE>   10

                  COMPANY: As defined in the introductory paragraph of this
         Agreement. For purposes of this Agreement, all references to the
         Company shall be deemed to include any successor entity or transferee.

                  EXCHANGE ACT: The Securities Exchange Act of 1934, or any
         similar federal statute, and the rules and regulations of the
         Commission thereunder, all as the same shall be in effect from time to
         time. Reference to a particular section of the Exchange Act shall
         include a reference to the comparable section, if any, of any other
         similar federal statute.

                  FRIENDS OF LIME ROCK: As defined in the introductory paragraph
         of this Agreement.

                  HOLDER: Friends of Lime Rock, Riverside and any Person to
         which either of such Persons transfers Registrable Securities.

                  PERSON: A corporation, an association, a partnership, a
         business, an individual, a governmental or political subdivision
         thereof or a governmental agency.

                  REGISTRABLE SECURITIES: Any of the Company's Common Stock
         issued to the Holders pursuant to the Contribution Agreement, and any
         securities issued to issuable with respect to any such Common Stock by
         way of distribution or in connection with any reorganization,
         recapitalization, merger, consolidation or otherwise. As to any
         particular Registrable Securities, once issued such securities shall
         cease to be Registrable Securities when (a) a registration statement
         with respect to the sale of such securities shall have become effective
         under the Securities Act and such securities shall have been disposed
         of in accordance with such registration statement, (b) they shall have
         been distributed to the public pursuant to Rule 144 (or any successor
         provision) under the Securities Act, (c) with respect to any Holder,
         the Registrable Securities held by such Holder represent less than 1%
         of the then issued and outstanding shares of Common Stock, such Holder
         is not an "affiliate" of the Company, and such shares may be sold
         freely by such Holder without restriction under Rule 144(k) (or any
         successor provision), or (d) they shall have ceased to be outstanding.

                  REGISTRATION EXPENSES: All expenses incident to the Company's
         performance of or compliance with Section 2.1 or 2.2, including,
         without limitation, all salaries of Company personnel or general
         overhead expenses of the Company, auditing fees, premiums or other
         expenses relating to liability insurance required by underwriters of
         the Company, or other expenses for the preparation of financial
         statements, all registration, filing and National Association of
         Securities Dealers fees, all fees and expenses of complying with
         securities or blue sky laws, all word processing, duplicating and
         printing expenses, messenger and delivery expenses, the fees and
         disbursements of counsel for the Company and of its independent public
         accountants, including the expenses of any special audits or "cold
         comfort" letters required by or incident to such performance and
         compliance, the reasonable fees and disbursements of not more than one
         special counsel selected by the holders of a majority of the
         Registrable Securities included in such

                                      -10-
<PAGE>   11

         registration, and any fees and disbursements of underwriters
         customarily paid by issuers or sellers of securities, but excluding
         Selling Expenses.

                  RIVERSIDE: As defined in the introductory paragraph of this
         Agreement.

                  SECURITIES ACT: The Securities Act of 1933, or any similar
         federal statute, and the rules and regulations of the Commission
         thereunder, all as of the same shall be in effect from time to time.
         References to a particular section of the Securities Act shall include
         a reference to the comparable section, if any, of any other similar
         federal statute.

                  SELLING EXPENSES: Underwriting discounts and commissions and
         stock transfer taxes relating to securities registered by the Holders.

         4. Rule 144: The Company will file the reports required to be filed by
it under the Securities Act and the Exchange Act and the rules and regulations
adopted by the Commission thereunder and will take such further action as the
Holders may reasonably request, all to the extent required from time to time to
enable the Holders to sell Registrable Securities without registration under the
Securities Act within the limitation of the exemptions provided by (a) Rule 144
under the Securities Act, as such rule may be amended from time to time or (b)
any similar rule or regulation hereafter adopted by the Commission. Upon the
request of any Holder, the Company will deliver to such holder a written
statement as to whether it has complied with such requirements. After any sale
of Registrable Securities pursuant to this Section 4, the Company will, to the
extent allowed by law, cause any restrictive legends to be removed and any
transfer restrictions to be rescinded with respect to such Registrable
Securities.

         5. Amendments and Waivers. This Agreement may be amended and the
Company may take any action herein prohibited or omit to perform any act herein
required to be performed by it, only if the Company shall have obtained the
written consent to such amendment, action or omission to act, of the Holders of
a majority of the Registrable Securities. The Holders of any Registrable
Securities at the time or thereafter outstanding shall be bound by any consent
authorized by this Section 5, whether or not such Registrable Securities shall
have been marked to indicate such consent.

         6. Nominees for Beneficial Holders. In the event that any Registrable
Securities are held by a nominee for the beneficial owner thereof, the
beneficial owner thereof may, at its election, be treated as the holder of such
Registrable Securities for purposes of this Agreement. If the beneficial owner
of any Registrable Securities so elects, the Company may require assurances
reasonably satisfactory to it of such owner's beneficial ownership of such
Registrable Securities.

         7. Notices. All communications provided for hereunder shall be sent by
first-class mail and (a) if addressed to Riverside or Friends of Lime Rock, at
Lime Rock Management LP, 518 Riverside Avenue, Bldg. 2, 2nd Floor, Westport,
Connecticut 06880, Attn: John Reynolds, (b) if addressed to any other Holder, to
the address specified by such Holder upon its receipt of Registrable Securities,
or (c) if addressed to the Company, at 401 Whitney Avenue, Suite 400, Gretna,
Louisiana 70056-2596, Attn: William J. Blackwell or at such other address, or to
the attention of such other officer, as the Company shall have furnished to the
Holders at the time

                                      -11-
<PAGE>   12

outstanding; provided, however, that any such communication to the Company may
also, at the option of any Holder, be either delivered to the Company at its
address set forth above or to any officer of the Company.

         8. Assignment. This Agreement shall be binding upon and inure to the
benefit of and be enforceable by the parties hereto and their respective
successors and assigns. In addition, and whether or not any express assignment
shall have been made, the provisions of this Agreement that are for the benefit
of any Holder shall also be for the benefit of and enforceable by any subsequent
holder of any Registrable Securities, subject to the provisions respecting the
minimum numbers or percentages of shares of Registrable Securities required in
order to be entitled to certain rights, or take certain actions, contained
herein.

         9. Descriptive Headings. The descriptive headings of the several
sections and paragraphs of this Agreement are inserted for reference only and
shall not limit or otherwise affect the meaning hereof.

         10. Specific Performance. The parties hereto recognize and agree that
money damages may be insufficient to compensate the Holders for breaches by the
Company of the terms hereof and, consequently, that the equitable remedy of
specific performance of the terms hereof will be available in the event of any
such breach.

         11. Governing Law. THIS AGREEMENT SHALL BE CONSTRUED AND ENFORCED IN
ACCORDANCE WITH, AND THE RIGHTS OF THE PARTIES SHALL BE GOVERNED BY, THE LAWS OF
THE STATE OF DELAWARE WITHOUT REGARD TO PRINCIPLES OF CONFLICTS OF LAWS.

         12. Counterparts. This Agreement may be executed simultaneously in any
number of counterparts, each of which shall be deemed an original, but all such
counterparts shall together constitute one and the same instrument.

         13. Other Agreements. For as long as there are any Registrable
Securities outstanding, the Company will not grant any registration rights more
favorable to the holder thereof than those granted to the Holders by this
Agreement, or that reduce or conflict with or are inconsistent with the rights
given to the Holders, in each case without the consent of the Holders of a
majority of the Registrable Securities.

                                      -12-
<PAGE>   13


         IN WITNESS WHEREOF, the parties have caused this Agreement to be
executed and delivered by their respective officers thereunto duly authorized as
of the date first above written.

                                    COMPANY:

                                    TORCH OFFSHORE, INC.

                                    By:/s/ Lyle Stockstill
                                    Name:
                                         ---------------------------------------
                                    Title:
                                          --------------------------------------


                                    FRIENDS OF LIME ROCK LP
                                    By:  Lime Rock Partners LLC
                                         its general partner


                                    By:/s/ John Reynolds

                                    Name:  John Reynolds
                                    Title: Managing Member


                                    RIVERSIDE INVESTMENTS LLC
                                    By:  The Beacon Group Investment Fund II,
                                         its sole member
                                    By:  Beacon Energy Investors II, L.P.,
                                         its general partner


                                    By:/s/ Bryan Martin

                                    Name:  Bryan Martin
                                    Title: Authorized Signatory


                                      -13-