The "best" is sort of hard to get to without carlifying a lot of factors.
First off would be budget...I would argue that all things being equal the Cooper LVT is prbably the nicest little rimfire out there but that's no budget gun.

Over all for rabbits and squirrels, the Marlin 980 V (NOT the 880SQ) is a pretty nice little gun and at a budget.

The 77/22 is nice and for a little bit more you can send the gun to Randy at CPC and really get it tweaked. Also the CZ series rifles do a nice job on a budget...and oddly the IZH Biathlon Basci is a novel shooting bolt action that has a GREAT trigger out of the box and is accurate beyond what you would ever expect for such an inexpensive rifle. One downside is that is can only use a scope.