How can I find out which special thieves guild quests I have completed?

Solution 1:

First off, you can tell how many you've done by looking around the Ragged Flagon; each quest adds a merchant to a stall. So if you count the merchants, aside from Tonilia, that should let you know the count.

The quests are:

Silver Lining (Markarth) - TGTQ01
Clear out Pinewatch of bandits and retrieve a mold. Endon will also become a fence.

The Dainty Sload (Solitude) - TGTQ02
Plant drugs on a ship captain.

Imitation Amnesty (Whiterun) - TGTQ03
Help remove records of a prisoner held in Dragonsreach before he's fingered for a murder in Solitude.

Summerset Shadows (Windhelm) - TGTQ04
Exact revenge for a murder committed by an upstart thievery ring holed up in Uttering Hills Cave. Niranye becomes a fence and can now buy anything.

If you have the PC version, you can check the status of those quest IDs above using something like GetStage TGTQ01.