This is taken from the prima guide and i thought i would like to share it since some of them are literally a pain in the butt to find.
I will take screenshots of all the locations when i have time to better describe where they are, but this will have to suffice for now, remember all the unusual gems are never locked inside something they are always on display keep that in mind when you look for each gem.
Remember to do all theese before you start searching for the specific gems.
Prerequisites:
[1/24] Proudspire Manor purchased
[3/24] During or after Main Quest: Diplomatic Immunity
[21/24] During or after Dark Brotherhood Quest: With Friends Like These... OR Dark Brotherhood Quest: Destroy the Dark Brotherhood
[1/24] Solitude (Proudspire Manor) = In the master bedroom of Proudspire Manor.
Link[2/24] Solitude (Blue Palace) = On a shelf in Jarl Elisif the Fair's quarters.
Link[3/24] Thalmor Embassy = In Elenwen's Solar in the Thalmor Embassy, in one of the bedrooms.
Link[4/24] Dainty Sload = On a small table in the Captain’s Quarters of the Dainty Sload, moored near Solitude.
Link[5/24] College of Winterhold = On a shelf in the Arch-Mage’s Quarters.
Link[6/24] Yngvild = In the Throne Room area of Yngvild, in the chamber behind the throne.
Link[7/24] Hob’s Fall Cave = In the necromancers' sleeping area.
Link[8/24] Markarth (Treasury House) = On a nightstand next to the bed in the master bedroom.
Link[9/24] Markarth (Understone Keep) = On a table in a locked side room of the Dwemer Museum.
Link[10/24] Dead Crone Rock = On a makeshift altar at Dead Crone Rock.
Link[11/24] Whiterun (Jorrvaskr) = In Kodlak Whitemane’s bedroom.
Link[12/24] Whiterun (Hall of the Dead) = In one of the wall crypts at the foot of a skeleton.
Link[13/24] Whiterun (Dragonsreach) = In the Jarl's bedroom area of the Jarl's Quarters.
Link[14/24] Rannveig’s Fast = On a table near the watery prison within Rannveig’s Fast.
Link[15/24] Fellglow Keep = On a counter in the workroom at the top of the front foyer.
Link[16/24] Windhelm (House of Clan Shatter-Shield) = In a bedroom of the Shatter-Shield home.
Link[17/24] Windhelm (Palace of the Kings) = On a table in Wuunferth the Unliving's quarters.
Link[18/24] Stony Creek Cave = In the Bandit Wizard's cavern in Stony Creek Cave.
Link[19/24] Ansilvund = Near Fjori’s ghost in the burial chambers of Ansilvund.
Link[20/24] Sunderstone Gorge = On the altar in front of the Word Wall.
Link[21/24] Dark Brotherhood Sanctuary = On the dresser in Astrid’s room.
Link[22/24] Pinewatch = In a locked treasure room in the Pinewatch Bandit Sanctuary.
Link[23/24] Riften (Mistveil Keep) = In the Jarl’s Chambers in Mistveil Keep, on the bedside table.
Link[24/24] Riften (Black-Briar Lodge) In the upstairs master bedroom of Black-Briar Lodge.
Link After finding all 24 Vex sends you off to find the Crown of Barenziah for the stones, after you hand the Crown in you recieve a perk called Prowler's Profit which vastly increases the chances of finding gems while scavenging dungeons.
The completed crown with gems is in the Guild behind Mercer’s desk once you complete the quest.
[Edited by sauced, 11/19/2011 5:36:28 PM]