I would like to clarify what Jillybean says about the .17hmr as it could be misunderstood the way she has written it. It is a very useful calibre, but you should note that it should only be used on squirrels if they are on the ground with a safe backstop. It should NEVER be fired at a squirrel in a tree, even if it does have the trunk behind it. If you were to miss the squirrel and the tree, the bullet could travel several miles, and what goes up must come down. What you don't want is someone to be in the way when it does come down. The same goes for the .22LR, it is not as fast or powerful as the .17hmr but it will still travel up to a mile or so. Remember the onus is on you to be sure that the shot you take is safe, you are responsible for where that projectile ends up.
The only guns that can be safely discharged upwards towards the tree canopy are shotguns and air rifles. I would be uneasy using a FAC rated air rifle (over the 12ft/lb license free limit) for this purpose as well, depending upon the power rating of it.
FWIW I don't think anybody was advocating hunting with an air pistol, but using one as a despatch tool for a trapped animal. I agree that it is not up to the job and would prefer a more capable tool. I am firmly of the belief that if you are to take the life of a sentient creature then your priority is to do it with the least suffering possible. For that reason I don't like live trapping, although it has to be done in some circumstances.
Sorry for such a preachy first post, but I get all twitchy when I see anything about firearms use that could be misunderstood, or even worse, misrepresented.
I am happy to answer any questions people have about use of firearms, if I don't know the answer, I know plenty of people who will!!