<PAGE>   1
      As filed with the Securities and Exchange Commission on June 16, 1998
                                            Registration Statement No. 333-
================================================================================



                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                              -------------------

                                    FORM S-8
                             REGISTRATION STATEMENT
                                      UNDER
                           THE SECURITIES ACT OF 1933

                              -------------------

                           BRIGHAM EXPLORATION COMPANY
             (Exact name of registrant as specified in its charter)



                 DELAWARE                              75-2692967
       (State or other jurisdiction        (I.R.S. Employer Identification No.)
    of incorporation or organization)        

  6300 BRIDGE POINT PARKWAY, BUILDING 2, 
        SUITE 500, AUSTIN, TEXAS                          78730
(Address of principal executive offices)                (Zip Code)

                              -------------------

                               1997 INCENTIVE PLAN
                         OF BRIGHAM EXPLORATION COMPANY
                            (Full title of the Plan)

                              -------------------

                  Ben M. Brigham                              Copy to:         
        Chief Executive Officer, President                 Joe Dannenmaier     
             and Chairman of the Board                   Thompson & Knight,    
            Brigham Exploration Company              A Professional Corporation
 6300 Bridge Point Parkway, Building 2, Suite 500        1700 Pacific Avenue   
                Austin, Texas 78730                          Suite 3300        
      (Name and address of agent for service)            Dallas, Texas 75201   
                                                           (214) 969-1700      
                  (512) 427-3300                    
           (Telephone number, including
         area code, of agent for service)

                              -------------------

                         CALCULATION OF REGISTRATION FEE

<TABLE>
<CAPTION>
===============================================================================================================================
 Title of securities to be     Amount to be     Proposed maximum offering        Proposed maximum             Amount of
         registered           registered(1)         price per share(2)     aggregate offering price(2)     registration fee
-------------------------------------------------------------------------------------------------------------------------------
<S>                             <C>                       <C>                      <C>                          <C>   
Common Stock par value          1,588,169                 $10.91                   $17,326,923                  $5,111
$.01 per share .............      shares
===============================================================================================================================
</TABLE>

(1)  Pursuant to Rule 416 under the Securities Act of 1933, shares issuable upon
     any stock split, stock dividend or similar transaction with respect to
     these shares are also being registered hereunder.

(2)  Estimated solely for the purpose of determining the registration fee
     pursuant to Rule 457(h).



================================================================================
<PAGE>   2



                                     PART I

              INFORMATION REQUIRED IN THE SECTION 10(a) PROSPECTUS

Item 1.      Plan Information.*

Item 2.      Registrant Information and Employee Plan Annual Information.*

-------------

*    Information required by Part I to be contained in the Section 10(a)
     prospectus is omitted from this Registration Statement in accordance with
     Rule 428 under the Securities Act of 1933 and the Note to Part I of Form
     S-8.





<PAGE>   3


                                     PART II

               INFORMATION REQUIRED IN THE REGISTRATION STATEMENT

Item 3.      Incorporation of Documents by Reference.

       The following documents filed by the Registrant with the Securities and
Exchange Commission are incorporated by reference in this Registration
Statement:

       (1)    The Company's Quarterly Report on Form 10-Q for the fiscal quarter
              ended March 31, 1998.

       (2)    The Company's Annual Report on Form 10-K for the fiscal year ended
              December 31, 1997.

       (3)    The Company's Current Report on Form 8-K (Date of Event: November
              12, 1997) dated January 23, 1998.

       (4)    All other reports filed by the Company pursuant to Section 13(a)
              or 15(d) of the Exchange Act since December 31, 1997.

       (5)    The description of the Common Stock contained in the Company's
              Registration Statement on Form 8-A (Registration Statement No.
              000-22433), including any amendment or report filed for the
              purpose of updating such description.

       In addition, all documents filed by the Registrant pursuant to Sections
13(a), 13(c), 14 and 15(d) of the Securities Exchange Act of 1934 subsequent to
the date of this Registration Statement and prior to the filing of a
post-effective amendment which indicates that all securities offered have been
sold or which deregisters all securities then remaining unsold shall be deemed
to be incorporated by reference in this Registration Statement and to be a part
hereof from the date of filing of such documents.


Item 4.      Description of Securities.

       Not applicable.


Item 5.      Interests of Named Experts and Counsel.

       Not Applicable.


Item 6.      Indemnification of Directors and Officers.

       In accordance with Section 102(b)(7) of the Delaware General Corporation
Law ("DGCL"), the Company's Certificate of Incorporation includes a provision
that, to the fullest extent permitted by law, eliminates the personal liability
of members of its Board of Directors to the Company or its stockholders for
monetary damages for breach of fiduciary duty as a director. Such provision does
not eliminate or limit the liability of a director (1) for any breach of a
director's duty of loyalty to the Company or its stockholders, (2) for acts or
omissions not in good faith or which involve intentional misconduct or a knowing
violation of a law, (3) for paying an unlawful dividend or approving an illegal
stock repurchase (as provided in Section 174 of the DGCL) or (4) for any
transaction from which the director derived an improper personal benefit.

       Under Section 145 of the DGCL, the Registrant has the power to indemnify
any person who was or is a party or is threatened to be made a party to any
threatened, pending or completed action, suit or proceeding (other than an
action by or in the right of the corporation) by reason of the fact that the
person is or was a director, officer, employee or agent of any corporation,
partnership, joint venture, trust or other enterprise, reasonably incurred in
connection with such action,




                                      II-1

<PAGE>   4




suit or proceeding. The power to indemnify applies only if the person acted in
good faith and in a manner the person reasonably believed to be in or not
opposed to the best interests of the corporation, and with respect to any
criminal action or proceeding, had no reasonable cause to believe the person's
conduct was unlawful.

       In the case of an action by or in the right of the Registrant, no
indemnification may be made with respect to any claim, issue or matter as to
which such person shall have been adjudged to be liable to the Registrant unless
and only to the extent that the court of chancery or the court in which such
action or suit was brought shall determine that despite the adjudication of
liability such person is fairly and reasonably entitled to indemnity for such
expenses which the court shall deem proper. Section 145 of the DGCL further
provides that to the extent a director or officer of the Registrant has been
successful in the defense of any action, suit or proceeding referred to above or
in the defense of any claim, issue or matter therein, that person shall be
indemnified against expenses (including attorney's fees) actually and reasonably
incurred in connection therewith.

       The Registrant also has the power to purchase and maintain insurance on
behalf of any person covering any liability incurred in that person's capacity
as a director, officer, employee or agent of the corporation, or arising out of
that person's status as such, whether or not the corporation would have the
power to indemnify against the liability.

       The Certificate of Incorporation and Bylaws provide that the Registrant
will indemnify its officers and directors and former officers and directors
against any expenses, judgments or settlement payments sustained or paid by such
persons as a result of having acted as an officer or director of the Registrant,
or, at the request of the Registrant, as an officer, director, agent or employee
of another business entity. The Certificate of Incorporation and Bylaws further
provide that the Registrant may, by action of its Board of Directors, provide
indemnification to employees and agents of the Registrant, individually or as a
group, with the same scope and effect as the indemnification of directors and
officers.

       The Company has entered into an Indemnity Agreement with each of its
executive officers, which provides for the indemnification in certain instances
against liability and expenses incurred in connection with proceedings brought
by or in the right of the Company or by third parties by reason of a person
serving as an officer or director of the Company.

Item 7.      Exemption from Registration Claimed.

       Not Applicable.


Item 8.      Exhibits.

       The following documents are filed as exhibits to this Registration
Statement:

       4.1    1997 Incentive Plan of Brigham Exploration Company (filed as
              Exhibit 10.9 to the Registrant's Registration Statement on Form
              S-1 (Registration No. 333-22491)) and incorporated herein by
              reference.

       5.1    Opinion of Thompson & Knight, A Professional Corporation.

       23.1   Consent of Thompson & Knight, A Professional Corporation (included
              in the opinion of Thompson & Knight, P.C. filed herewith as
              Exhibit 5.1).

       23.2   Consent of Price Waterhouse LLP.

       24.1   Power of Attorney (included on signature page of this Registration
              Statement).




                                      II-2

<PAGE>   5




Item 9.      Undertakings.

       The Registrant hereby undertakes:

             (1) To file, during any period in which offers or sales are being
       made, a post-effective amendment to this Registration Statement:

                     (i) To include any prospectus required by Section 10(a)(3)
              of the Securities Act of 1933;

                     (ii) To reflect in the prospectus any facts or events
              arising after the effective date of this Registration Statement
              (or the most recent post-effective amendment thereof) which,
              individually or in the aggregate, represent a fundamental change
              in the information set forth in this Registration Statement.
              Notwithstanding the foregoing, any increase or decrease in volume
              of securities offered (if the total dollar value of securities
              offered would not exceed that which was registered) and any
              deviation from the low or high end of the estimated maximum
              offering range may be reflected in the form of prospectus filed
              with the Commission pursuant to Rule 424(b) if, in the aggregate,
              the changes in volume and price represent no more than a 20%
              change in the maximum aggregate offering price set forth in the
              "Calculation of Registration Fee" table in the effective
              registration statement; and

                     (iii) To include any material information with respect to
              the plan of distribution not previously disclosed in this
              Registration Statement or any material change to such information
              in this Registration Statement;

       provided, however, that paragraphs (1)(i) and (1)(ii) do not apply if the
       information required to be included in a post-effective amendment by
       those paragraphs is contained in periodic reports filed by the Registrant
       pursuant to Section 13 or Section 15(d) of the Securities Exchange Act of
       1934 that are incorporated by reference in this Registration Statement.

             (2) That, for the purpose of determining any liability under the
       Securities Act of 1933, each such post-effective amendment shall be
       deemed to be a new registration statement relating to the securities
       offered therein, and the offering of such securities at that time shall
       be deemed to be the initial bona fide offering thereof.

             (3) To remove from registration by means of a post-effective
       amendment any of the securities being registered which remain unsold at
       the termination of the offering.

             (4) That, for purposes of determining any liability under the
       Securities Act of 1933, each filing of the Registrant's annual report
       pursuant to Section 13(a) or Section 15(d) of the Securities Exchange Act
       of 1934 that is incorporated by reference in this Registration Statement
       shall be deemed to be a new registration statement relating to the
       securities offered therein, and the offering of such securities at that
       time shall be deemed to be the initial bona fide offering thereof.

             (5) Insofar as indemnification for liabilities arising under the
       Securities Act of 1933 may be permitted to directors, officers and
       controlling persons of the Registrant pursuant to the foregoing
       provisions, or otherwise, the Registrant has been advised that in the
       opinion of the Securities and Exchange Commission such indemnification is
       against public policy as expressed in the Act and is, therefore,
       unenforceable. In the event that a claim for indemnification against such
       liabilities (other than the payment by the Registrant of expenses
       incurred or paid by a director, officer or controlling person of the
       Registrant in the successful defense of any action, suit or proceeding)
       is asserted by such director, officer or controlling person in connection
       with the securities being registered, the Registrant will, unless in the
       opinion of its counsel the matter has been settled by controlling
       precedent, submit to a court of appropriate jurisdiction the question
       whether such indemnification by it is against public policy as expressed
       in the Act and will be governed by the final adjudication of such issue.



                                      II-3

<PAGE>   6


                                   SIGNATURES

       Pursuant to the requirements of the Securities Act of 1933, the
Registrant certifies that it has reasonable grounds to believe that it meets all
of the requirements for filing on Form S-8 and has duly caused this Registration
Statement to be signed on its behalf by the undersigned, thereunto duly
authorized, in the City of Austin and State of Texas on the 28th day of May,
1998.

                                          BRIGHAM EXPLORATION COMPANY


                                          By:   /s/ Ben M. Brigham
                                                -------------------------------
                                                Ben M. Brigham, Chief Executive
                                                Officer, President and Chairman
                                                of the Board


                                POWER OF ATTORNEY

       KNOW ALL MEN BY THESE PRESENTS, that the undersigned directors and
officers of Brigham Exploration Company, a Delaware corporation, which is filing
a Registration Statement on Form S-8 with the Securities and Exchange Commission
under the provisions of the Securities Act of 1933, as amended, hereby
constitute and appoint Ben M. Brigham and Anne L. Brigham, and each of them, his
true and lawful attorneys-in-fact and agents, with full power of substitution
and resubstitution, for him and in his name, place and stead, in any and all
capacities, to sign such Registration Statement and any or all amendments to the
Registration Statement, and all other documents in connection therewith to be
filed with the Securities and Exchange Commission, granting unto said
attorneys-in-fact and agents, and each of them, full power and authority to do
and perform each and every act and thing requisite and necessary to be done in
and about the premises, as fully to all intents and purposes as he might or
could do in person, hereby ratifying and confirming all that said
attorneys-in-fact as agents or any of them, or their substitute or substitutes,
may lawfully do or cause to be done by virtue hereof.

Pursuant to the requirements of the Securities Act of 1933, this Registration
Statement has been signed by the following persons in the capacities and on the
dates indicated.

<TABLE>
<CAPTION>
             SIGNATURE                                    TITLE                               DATE
             ---------                                    -----                               ----
<S>                                          <C>                                          <C>
        /s/ Ben M. Brigham                   Chief Executive Officer,                     May 28, 1998
--------------------------------------       President and Chairman of 
           Ben M. Brigham                    the Board (principal      
                                             executive officer)        
                                             

       /s/ Anne L. Brigham                   Executive Vice President                     May 28, 1998
--------------------------------------       and Director
          Anne L. Brigham                    


        /s/ Jon L. Glass                     Vice President -                             May 28, 1998
--------------------------------------       Exploration and Director
          Jon L. Glass                       


        /s/ Craig M. Fleming                 Chief Financial Officer                      May 28, 1998
--------------------------------------       (principal financial and
          Craig M. Fleming                   accounting officer)
                                             
</TABLE>





                                      II-4

<PAGE>   7




<TABLE>
<CAPTION>
           SIGNATURE                         TITLE               DATE
           ---------                         -----               ----
<S>                                        <C>               <C>
      /s/ Harold D. Carter                 Director          May 28, 1998
--------------------------------------
        Harold D. Carter


     /s/ Alexis M. Cranberg                Director          May 28, 1998
--------------------------------------
       Alexis M. Cranberg


      /s/ Gary J. Milavec                  Director          May 28, 1998
--------------------------------------
        Gary J. Milavec


    /s/ Stephen P. Reynolds                Director          May 28, 1998
--------------------------------------
       Stephen P. Reynolds
</TABLE>





                                      II-5

<PAGE>   8




                                INDEX TO EXHIBITS

<TABLE>
<CAPTION>
Exhibit Number               Exhibit
--------------               -------
<S>           <C>
    4.1       1997 Incentive Plan of Brigham Exploration Company (filed as
              Exhibit 10.9 to the Registrant's Registration Statement on Form
              S-1 (Registration Statement No. 333-22491)) and incorporated
              herein by reference.

    5.1       Opinion of Thompson & Knight, A Professional Corporation.

   23.1       Consent of Thompson & Knight, A Professional Corporation (included
              in the opinion of Thompson & Knight, P.C. filed herewith as
              Exhibit 5.1).

   23.2       Consent of Price Waterhouse LLP.

   24.1       Power of Attorney (included on signature page of this Registration
              Statement).
</TABLE>