Rabbits circle around, and come back to a spot very close to where you jumped them. You can chase them in circles, all day, if you're stubborn and not paying attention. Sometimes the circle is 10 yards in diameter; sometimes it's half a mile. Unless you catch them foraging very
far from their home, it's always a circle.
They're always watching. The rabbit knows you're there, long before you see it. Being able to spot the ears, eyes, or mouth is very helpful in getting the upper hand. Otherwise, you'll be waiting for them to provide a shot opportunity after they jump.
Rabbits will often wait until the very last moment, before jumping. I've actually stepped on
cottontails and snowshoes that waited until my boot was coming down on them, before they tried to run. (Sagebrush plateaus.) The closer they are to their warren (burrow) or form (above ground 'nest'), the longer they'll wait before giving up their position.
They like to run, pause, look back, and run again. If you jump a rabbit, stay still (or move minimally, to keep an eye on it). It should stop, and give you an over-the-shoulder glance. That's your last chance to turn it into supper, before it decides you're a real threat. Once it decides to run again, you won't get many shot opportunities. (Revert to Lesson #1)
If you don't have a shotgun, the odds of making a successful shot on a running rabbit are fairly low. It can be done, and I'm sure there are some excellent hunters that make it look easy; but I've never met one.
During the winter, they're 50% hair. Your best bets are head shots or center of chest (with meat loss). If you aren't precise with your shots, you'll just be giving them a haircut.
My personal preferences vary, depending on the species, terrain, vegetation, and time of year.
Generally, I go for the .22 WMR for areas that will provide longer shots (≥50 yards).
In areas where I'm likely to jump or spot them up close, I usually carry a handgun. But... a .22 rifle is not unheard of.
I almost never take a shotgun on a rabbit hunt. The animals are always too close (and will be blow to pieces), or too far away (and it won't be an ethical, fatal hit).
By species, I would say it usually breaks down like this:
Snowshoe - whatever I have in my hand (I don't actively hunt them - opportunistic kills, only). .22 LR would usually be sufficient.
Cottontail - .22 LR / .22 WMR
Jack - .22 WMR / anything centerfire (we get some big jacks out here