With the introduction of the analog stick and Rumble Pak, the N64 started some de facto standards.
Project64 is probably the most popular Nintendo64 emulator currently available. With excellent game support and very little bugs. There is also an unofficial version with netplay support and in our opinion project64 is the best emulator you could use.
Before the release of the source code for UltraHLE there was a hacked version that attempted to improve the emulator and actually resulted in a pretty decent N64 emulator. Worth trying as a secondary to Project64.
Mupen64++ is a version of Mupen64 built off of the original source code with added Kaillera support and features. This emualtor has a high compatibility and generally speaking isn't a bad choice.
As the name suggests 1964 is a N64 emulator also with a high compatibility. Probably in the same league as project64 but requires a higher minimum system requirement.