Between the only handheld consoles that count - PSP and NDS, PSP has the best emulation stability due to the fact that it's just pure awesome power in a box. That said, it's also limited too - it's far more complicated to get CFW running on PSP to run above homebrew/ISO/CSO/emulators, but they've got a number of "noobinstallers" and guides to get people up and running.
From what I've heard, PS1 and down all work fine (although some PS1 games need a certain version of POPSLoader to run better), N64 (with Daedalus, currently version R13 with R14 on the way) has a much lower compatability rate than PS1, but they've managed to get some games working pretty well (such as Mario Kart and Super Smash Bros), and the DC emulation project was scrapped because there's too little demand for it and the PSP isn't powerful enough to run most DC games (that use 128bit graphics). Fortunately, most of the DC games that mattered have already been ported out to other consoles.
EDIT: Some games run at a playable speed with sound off, some work with sound, but some games like Zelda: OoT become "artifacted" (don't ask me what that means - I've yet to receive an answer) and are unplayable. Basically, the main obstacles are speed (w/ or w/o sound) and graphics (64bit...).