Head shot w/ a .22 if they're not running, don't ruin any meat that way...Pellet gun isn't powerful enough for a clean kill.
But, to my knowledge they don't usually dig holes (although they'll live in existing ones), except when they make a nest, which is a really shallow hole. Used to have a bunch around here, never hurt my yard a bit, they'd just come out and "graze". Haven't seen any in some time, think the foxes (which we have alot of) and even coyotes (which are rapidly taking over) have thinned the herd. I just leave 'em alone, as I don't shoot anything that I don't eat (and I don't much care for rabbit) or is causing me trouble.
If you have holes in the yard, suspect woodchucks, or if smaller, moles, both can be real diggers.