I use ePSXe for PS1, PCSX2 for PS2, PPSSPP for PSP.
I also use VBA for GBA/GBC/GB, but since building my new PC it seems like it can't handle the new resources? Not sure if this is possible, or if my version is just a million years old. But it always worked fine, now when I play the same games as I always used to, eg FF Tactics Advanced, it stutters, doesn't register button presses (to the point that it is unworkable with my controller, I HAVE to use my keyboard.), and the graphics are all wonky.
Gonna take a look at this BizHawk thing now too.
EDIT: I also use Project64 for N64.
UPDATE: Liking the Bizhawk so far. Loaded my GBA stuff right, and seems to register the controller better. But what BIOS should I use for PSX?