The ZSNES developers focused on speed over accuracy, so a hefty amount of games have bugs (and some just plain crash). In the current 1.51 version many problems still remain:
Compatibility is lower than any modern emulator due to low accuracy to real SNES system.
Games crashing - Der Langrisser (freezes after 2-3 hours), Super Mario RPG (freezes in many different points of the game), Star Ocean and many other games have freezing and crashing issues.
Bad sound emulation - A ton of games sound terrible on ZSNES; Earthworm Jim 2 has no sound effects.
Super FX not clocked correctly - Star Fox runs twice as fast as it should.
Transparency - Kirby Dreamland 3, Mecarobot Golf, Jurassic Park and any game that uses pseudo hi-res translucency won't display correctly.
These are just a few problems, there are a few more listed here, and here.
It is recommended to use Snes9x or RetroArch with the bsnes core(s) instead.
Many ROM hacks and ROM hacking tools were made using ZSNES as a testing platform because of its popularity. However, because of the inaccuracies of the emulator, these tools and hacks are broken on other, more accurate emulators and real hardware.
In 2011, byuu released bZSNES - a modification of his bsnes that includes the optional inaccuracies of ZSNES and a similar GUI. This version of bsnes can accurately emulate the ROM hacks made specifically for ZSNES, which means the only thing ZSNES had going for it can now be played on a much more accurate emulator.