PS2 controllers are popular since they have been around for a decade and the communication protocol is well known and documented (just google it, it's all over the internet).
Gamecube controllers are also good candidates since they require fewer pins ( I can't remember if its 1 or 2 ), but this is less well documented.
SuperNES and NES controllers I believe have also been "hacked" into but these are mostly novelty items since they are hard to obtain these days, and they don't have as many features as more modern videogame controllers.
Also, as far as I know, no "USB type" controller (Xbox, and I believe any of the newer ones, PS3, Xbox360, maybe even Dreamcast, Wii) has been interfaced with a microcontroller. It may be possible to use a bluetooth module to do this, but I haven't seen it yet. That said, I think the Wiimote's nunchuck still uses a serial interface, although I'm not sure.
My advice-stick with wired controllers unless you need it to be wireless because it is a lot harder to get it working. For some reason, at least with PS2 controllers, interfacing with a wired controller is really easy (works with 100% of wired controllers), but with wireless, it's more like 20%, at least from my experience. Also if you do decide on a controller to go with, instead of hacking the controller cable/receiver apart, just buy a controller "extension cord" off ebay and make a nice breadboard adapter.