Snes9x-Next is an optimized version of Snes9x 1.52 with speedhacks and is available as a libretro core for RetroArch and ZMZ. Has an option to overclock the SuperFX chip, making games like Star Fox run at a higher FPS.
Snes9xTYL(mecm)/cm Mod is a port of Snes9x 1.39 to PSP. Inaccuracies everywhere, especially with sound, but it's the best thing available for PSP for SNES emulation. Use the mecm build where possible, unless you use the PSP emulator on Vita, in which case use the cm build.
It is a very good SNES emulator, far superior to ZSNES. It has very high compatibility and is easy to use. It is not as accurate as higan, but that shouldn't be an issue for most people. The stable release is very old, so use the latest SVN build instead if possible. Also, it's recommended not to use the official site because the developers have lost access to it many years ago.
Problems and solutions
Snes9x stopped recognizing diagonals and joystick/R2/L2 input? Snes9x has issues with Xinput (such as with XInput Wrapper SCP). The solution is to edit the .conf file and add in (J0)POV Up Right, (J0)POV Dn Right, etc. for the diagonal inputs.