Recording on external drives

No, it is better to work on the local hard drive. Recording and playback require that the audio data is available as a continuous stream without interruption. USB in particular may have unexpected delays reading or writing that can cause dropouts and other errors in the data, which could even cause the application to freeze or crash. I would guess that a firewire drive would probably be ok, but I don’t have one so I’ve never tested that.

It’s OK to transfer files to external drives when you have finished working on them, but not that for Audacity (and ProTools) the audio data is separate from the project file. See here for how to safely move Audacity projects:

Note also, that after an Audacity project has been saved, Audacity uses the project “_data” folder for temporary data, not the default “temp” folder. If you wish to resume work on a project that you have copied to an external drive, it is best to either copy the AUP and its associated audio data back to the internal drive before you start work, or, open the project from the external drive and then save the project to an internal hard drive before you start work (so that the project that you are working with is on the local hard drive).

Tip: After moving or copying important projects - check that they still play ok all the way through.
Tip 2: Moving or copying ordinary audio files (such as WAV files) is easier and safer than moving / copying projects, because a WAV file is just one file, not thousands of little bits.