My 2002 A6 3.0 Quattro shakes when idling, and the RMS needle bounces around slightly. It also shakes when accelerating but is perfectly fine when coasting.

Has anyone managed to fix this problem? If so please tell me how.

 

This sounds like a vacuum leak given the irregular RPMs you mentioned. Doing a quick search for vacuum leaks on here or on Google will yield plenty of results to get you started.