We had the same problem. The lawn looked like a cocoa puff cereal truck accident ... rabbit doo everywhere. The kids couldn't even play out there, and anything we planted just became bunny food within 24 hours. When we came home and counted something like 18 rabbits on the lawn (I'm
not kidding) we finally declared war.
We now use a Benjamin Sheridan .22 Model 392 air rifle, topped with a BSA air rifle scope, 2x7, with an adjustable objective (a scope is very helpful). See
www.crosman.com for information about the rifle ... it is a pump-action. I rather like that, because it's only a one-shot affair anyway, and I don't have to feed it anything except pellets. I believe we purchased the scope from Cabela's. Rifle was about $110, and the scope was $60. You'll need to buy a scope mount adapter from Crosman ... another $10 or so, as I recall.
Whatever you buy, ask about the noise. My air rifle is quiet enough that it doesn't cause a stir ... I gather that some are pretty loud. IMHO, one problem about using a .22 is potential legal hassles. Explaining an air rifle is one thing, but regardless of reality, LEO's / judges may not care about the details of your .22 ammunition - it is a firearm, and that is a big issue in town for many folks.
For a quick kill, I suggest aiming at the eye, or just behind the eye. This drops them like a rock. Don't go for body shots ... take the head shot, so it's 'lights out' and humane as possible.
Funny story ... I thought I was the only fellow around here so motivated. Found out that a neighbor does the same thing, with an air rifle. Then my neighbor asked me if I had heard any loud noises lately ... I said no. He said that another neighbor, a sharp, older fellow around the corner, takes out rabbits with his 12 gauge shotgun! Now I feel a little under-gunned.
The part I didn't expect? Some here might laugh, but I honestly believe this has made me a better shot. No kidding.
Good luck with your rabbits. Believe me ... this is an effective solution. And, in many situations, perhaps the only practical means to deal with the problem.
Regards from AZ