Skyrim:No Stone Unturned

Quick Walkthrough
# Find one of the 24 "Unusual Gems" scattered throughout Skyrim.
# Become a full member of the Thieves Guild (if not one already).
# Have the gem appraised by Vex at The Ragged Flagon in Riften.
# Collect the remaining stones of a full set and return to Vex.
# Retrieve the Crown of Barenziah and return to Vex.

Detailed Walkthrough
thumb|right|An Unusual Gem
What Do We Have Here?
In your travels about Skyrim, you will come upon an ornate jewel (one of twenty-four spread across Skyrim) called an Unusual Gem. Subsequent actions after you acquire one of these gems depends on your membership status with the Thieves Guild.

Not a Member of the Thieves Guild
If you are not a member of the Thieves Guild, picking up the first Unusual Gem you encounter will activate the miscellaneous quest objective: "Bring the Unusual Gem to an appraiser". 

Depending on your past travels and interactions, following your objective marker will likely take you to Maul, a Nord thief in Riften. If you have not yet spoken with Maul, he will seek you out with some unsolicited advice. After the conversation, ask him what he knows about the gem. He will tell you that you need to show it to Vex of the Thieves Guild, to find out what the gem is worth. Maul will also tell you that to get to Vex, you must first become a member of the Thieves Guild.
If you press for information on the guild, you will be told to find Brynjolf in Riften's marketplace. You can find him as suggested, or in the Bee and Barb at night. Talking to him will activate the quest A Chance Arrangement, which is the first step towards becoming a member of the Guild. Once you have access to The Ragged Flagon, seek out Vex for her appraisal of the gem. She will tell you that it is one of twenty-four stones that were pried from the crown of Queen Barenziah by a thief in an attempt to cover his tracks, and that only the complete set of twenty-four stones would be worth anything.
If Maul does not tell you to find Brynjolf, you can seek him out yourself.

Already a Member of the Thieves Guild
If you are a full member of the Thieves Guild when you pick up the first stone, the quest is immediately activated with the objective: "Bring the Unusual Gem to Vex in the Thieves Guild". Find Vex in The Ragged Flagon. She will tell you that it is one of twenty-four stones that were pried from the crown of Queen Barenziah by a thief in an attempt to cover his tracks. She will further inform you that the stone is worthless on its own, but that if you can find the other twenty-three, the collection would be worth a substantial sum.

Finding the Gems
After showing the Unusual Gem to Vex, the gems are renamed "Stones of Barenziah," and will appear as such in your inventory when in your possession. You must now find all twenty-four stones. 

You'll need to complete some other tasks before you have access to all locations where the gems are to be found:
Begin the Dark Brotherhood questline or start the Destroy the Dark Brotherhood! quest.
Purchase Proudspire Manor in Solitude. (25,000 gold)
Join the College of Winterhold by completing First Lessons. You must also start Under Saarthal and progress at least to the point where Tolfdir tells you to speak with the Arch-Mage; until this point, the Arch-Mage's Quarters are locked behind an inaccessible door.

Aside from those, most other gems can actually be obtained at any point in the game. Many are located in places that you will be sent to during other quests, so you may want to consider getting the stone concurrently with the quest.

The stones are located in the following places:



  Will be marked as stolen unless you are of an appropriate faction rank. 

Recovering the Crown
thumb|Find the Crown within Tolvald's Cave
Show Vex the full set of Stones, and she will have you travel to Tolvald's Cave to retrieve the Crown of Barenziah. The cave is three layers deep, and crawling with Falmer, so be sure to prepare yourself. When approaching the crown, three hostile Dunmer Ghosts will appear and attack. After returning with the Crown, you will receive the permanent effect Prowler's Profit, which significantly increases the chance of finding gems in randomized containers.

Notes
If you enter Torvald's Cave without having spoken to Vex about having all the gems, the area where the crown is found will be blocked off by rubble. This will cause you to have to return to Vex and then travel all the way through the cave again to find the crown.
The gems that will be marked as stolen may say "steal" instead of "take" when acquiring them; in this case, they will still be marked as stolen and if you're caught taking them, you will be reported.
If your game is prior to patch 1.4 and you haven't recovered the stone from within the Thalmor Embassy, there are multiple workarounds detailed on the location page.
There are no quest markers for the stones, so unless you intend to find all the stones by chance, it may be best to print out a list of their locations and cross each one off as you find it.
It is impossible to remove this quest from your list without joining the Thieves Guild; if you don't want to do that, you may be better off ignoring any Unusual Gems you come across, as they have no other use in the game and cannot be removed from your inventory without Vex.
The stone in the Dark Brotherhood Sanctuary is not generated by the game until either of the related quest lines is started.
Text in the game code suggests that you were originally supposed to see Tonilia about the gems. The dialogue about buying the gems would have made much more sense in that context.

Bugs
 
 To fix this, type these console commands:  , followed by  .
 
 
 
 
 Type the following three commands into the console for a workaround:  ;  ;  . This will not update the journal entry with the correct number of gems found but it will unlock Vex's dialogue to proceed with the quest.
 
 
 
 
As noted in the text, this can be bypassed by simply joining the Thieves Guild and ignoring Maul.
 

Quest Stages