NES, Master System, SNES, Mega Drive (and Sega CD), PC-Engine (and PC-E CD), GB, GBC, GBA, Virtual Boy, Wonderswan, Wonderswan Color, Neo Geo Pocket, and probably another console I forget seem to work perfect or almost perfect on the Wii with the proper emulator (mGBA for GBA, and not VBA, for example). Thanks to the infra-red on the Wiimote you can even play NES / SNES lightgun games on modern TVs and get a rather close experience to the original (you probably won't miss using 6 batteries on the superscope, though).
There are others, like MSX, which can work, but most games require you to have a keyboard connected (as the original system had one) and Atari 2600/7800. I don't put much time into arcade stuff, so I can't comment.
As for N64 and PS1, some games do certainly work, but the compatibility is way lower and only a few are what we could say "fully playable". I remember beating Symphony of the Night and Digimon World 1 on it, before getting a "new" PS2. You can check the compatibility lists done here to see if there's a game you like and is playable enough, but don't get your hopes up.
Most of the Retroarch emulators work very nicely on Wii.
N64 and Playstation emulators (Wii64 and WiiSX) run okay.
Which version of Retroarch do you use on the Wii?