Whippets. This is a question with a definitively correct answer - They were bred for rabbit catching. Lots of dogs can catch rabbits, but nothing comes close to whippets.
----- Don't get me wrong - getting ANY dog is almost always a good investment, but getting a dog specifically for rabbit? Then Whippets are the most sensible choice.
The benefit of a whippet is that, essentially, it's a cat. It sleeps for 16-18 hours a day, doesn't smell, doesn't eat much, is very very clean and very very lazy. a 15 to 20 minute walk is about all they need because if they go much further they get tired and sit down and refuse to walk. They are also by far the gentlest of all the dog species - they make Labradors look aggressive, and are brilliantly safe with children. So - when you DON'T want to catch rabbits, you have a very easy dog to live with.
But... when they see a rabbit, they are the second fastest land animal on earth (after the cheetah) and will happily catch the rabbits as long as you like. Their only desires in life are: 1) sleep on the sofa and 2) catch rabbits. Then, when your done, they'll sleep for another 15 hours.
Beware though - they are astonishingly dumb - when they HAVE the rabbit, they often look a bit confused as to what to do next.
But - if you want to catch rabbits, using a dog isn't especially sensible or efficient - even a whippet will only consistently catch perhaps 1 in 6 or even 1 in 8. If you have a rabbit problem, then you need to be using ferrets - by far and away the most efficient way of catching rabbits - even more efficient than gas - and you get a lot of meat and furs as well.
If you have hares instead of rabbits, the hare has more stamina than a Whippet and will usually out-run it. In this case, either a greyhound or a saluki is the better option - both are a little slower, but have got more stamina. A greyhound is just like a large whippet in terms of temperament and exercise requirements - possibly they are even more lazy. A saluki is as gentle as a whippet, but their stamina is astonishing - right up their with Vizsla's in terms of endurance - if you are going to use them as a working dog that is fine, but as a pet at home, you simply can't give them enough exercise.
Steve ( owner of whippets and regular rabbit catcher)