The questions I find here are probably at least 50% kids thinking their under powered BB and sometimes pellet guns are ok to take pot shots at what ever little animal they have a lot of. You'll see a lot of those answered with the fact that it would be more humane clubbing them with the stock than shooting anything. The other popular one is dropping an anvil on them. You could get well in excess of 50 or even 100 ft-lbs from an anvil but less than 2 ft-lbs from your average BB gun.
I bought my newest airgun with the idea of large pest removal in mind. While it isn't urban,where I live is still suburban and discharge of firearms would be a problem legally. That rules out every other gun i own because they make a lot of noise. So far I'm winning the groundhog war and every kill has been as humane as I could make it. Not one has moved from the spot where it was shot. I grew up with different airguns and firearms so the airgun was a natural choice once I found out it was legal for hunting where I live. This perhaps covers another 25% or more questions using airguns for quiet pest removal. Few of them are after anything larger than squirrels so they really don't get into anything "high powered" as far as airguns go. Pest removal is covered under hunting regulations in most states. So technically pest removal is hunting. Sometimes you have to look for nuisance animals in the regulations and they are often open season if they are causing various problems. There is a few places that even have a bounty on some pests.
Probably less than 5% are from serious hunters. After all, if you can use a .22LR, most would chose that over an airgun in many circumstances. Even the price vs accuracy would favor firearms in almost all cases. The challenge of airgun hunting is you have to get much closer for a good clean kill. I've had to pass on shots that I could have easily pulled off with a .22LR, .17HMR or .22mag. Keep in mind that pest removal and hunting is virtually the same though pests tend to be much closer to houses and other buildings. Which is another reason for airguns over firearms.
By choosing a low powered (.177 1200fps) airgun, you can have plenty of power for anything up to squirrels. By choosing one of the magnum powered airguns you can go up to groundhogs, jack rabbits and even raccoons. These are all generally break barrel airguns. By choosing a PCP, you can get some impressive power that can exceed .22LR standards as well as larger calibers with even more power. There are airguns that are more than capable of taking down a deer though an equivalent firearm would be much cheaper.
Knowing what any particular airgun will do will require further investment in a chronograph and perhaps ballistics software. While any number of people can give ballpark numbers here, It doesn't quite illustrate the actual limits. Mostly though it is close enough with an added margin for overkill.
Shadow Wolf