Depends on the pest. I often use a .177 air pistol for attic rats - though I once crawled up there to discover the largest raccoon I ever saw and immediately felt completely unarmed with the thing. After briefly considering returning with a .357, I went downstairs, poured myself a drink, and carefully ignored any further noises from the attic.
Otherwise, I get a good deal of use from homemade shot cartridges in my .500 S&W. I fill them with #4 or #6, and they essentially duplicate the performance of a 20 gauge, taking care of snakes and rodents with little fuss and no danger to the neighbors.