My brother has actually made a pretty good sporting event out of rats... he uses a .22 cal air rifle (started with the spring, liked it so much he now has upgraded to one with an air cannister). He got the ones with the integral silencer, so they aren't loud.
Couple trips to Harbor Freight netted him a cheap surveillance camera system with IR capability, a cheap motion-detector light that he put IR bulbs in. Rigged it up so that the back yard will be hit with IR floodlights, completely dark to the naked eye but visible to the IR cameras. Also set up so a red lamp comes on inside whenever the motion lamp is triggered.
Found an older Sony camcorder that has an IR setting, and with a little hoseclamp and rigging, fixed it to the scope on the air rifle. That stays on a benchrest set up by a back window in the sunroom... the camcorder both functions as a night sight, and records the event as seen via the scope.
The light will go off, he will check the security camera (has it set up to view through an old TV). We get a bright look at the entire back yard, and the IR lights will reflect off the rat's eyes so you can spot critters pretty quick, even if in foliage. Then, it's sniper time.
When I first saw the setup. I thought it was pretty overdone, but he's taken a whole lot, and it's pretty darn fun to do so.
He might have $150 at the most invested in the entire setup minus the actual rifle.