Sparrows that is, the little suckers try to nest in my Bluebird boxes every year. Every year I kill about a dozen before they decide to try somewhere else.
I usually have to trap 5 to 7 house sparrows out of my bluebird boxes each spring before the bluebirds and tree swallows have a chance to nest. Last summer, I trapped sparrows from May through October in a walk-in style ground trap. Ended up with 127 for the season. They were mostly recently fledged young though. I caught very few adults in the ground trap.
I used a 28 gauge this AM, they're a little hard for me to hit flying with the CF. I don't think it would take much bullet with a 308 though, a 110 gr hollow point would probably be plenty.
Be aware that only english sparrows will nest in your bird boxes. The other kinds of sparrows leave the boxes alone. Best to just shoot the ones that are the problem and let the rest go.
I have picked quite a few off with 22 shorts, some long rifles, and once a 22 Hornet. The 22 birdshot shells from a handgun are plenty too, at close range. Once had a starling get stuck in the hole trying to get back out. It didn't end well for him.