Difference between countries, but here it is legal as long as all of the following criteria are met: a) the animal in question isn't endangered or specifically protected, b) if domesticated, that you have destruction rights for the animal (generally the case for pets), and c) it is killed in a "humane way" which is a rather long list of don'ts.

There was a woman in my town charged for animal abuse by trapping mice in a "wrong" way that didn't properly minimise suffering.

Depends where you live. Wild animals, we called local animal control about a opossum we'd caught in a cat trap (trying to catch a local stray cat) wanting to know what they wanted us to do with it. and they let us know about the felony we were about to admit to committing and did we want to shut the fuck up. Apparently we needed a license to trap wild animals. So telling your local animal control after the fact isn't in your best interests, but they could provide guidance before the fact. If you're in Washington State, you leave those (just put on the local endangered list) western gray squirrels the fuck alone.

Why are you interested in killing a squirrel?

I'm in the US. I would look under local small game rules. You should be able to find a list or table of critters considered game, and maybe specific rules on "nuisances". If it isn't called out as needing anything special then you're probably fine.

I know this would vary from locale to locale, but

But still you don't tell for which one you are asking?

Shit, if you do in my state I've broken the law hundreds of times. My friend and I used to go out to this bit of farm land specifically to shoot the squirrels, because the farmer paid a bounty on each one brought back as proof. It was like an IRL low-level gold/XP farming quest!

Georgia has a season for hunting squirrels. I believe you can kill feral hogs anytime but are supposed to use whatever is legal for the current hunting season. Most things have a season. Your state's DNR website should have lists.

In England it's illigal to keep a squirrel in captivity, but it's also illigal to release a captured squirrel back into the wild, so the only option is to (humanely) kill the poor fucker.

In Maine you can hunt red squirrel all year. Grey squirrel season is like September 1 to December 31. Can’t hunt greys outside those months there.

For species that are listed as nuisance animals, it can be legal to kill them without a hunting license and out of a hunting season. If there's evidence that the animal is causing significant harm, it can generally be killed.

There are multiple different rules in many locales. Here are the rule for MN (as of 20 years ago) The DNR has a season on them, so you have to have a small game license (no license needed if under 16), hunt in season, obey the maximum per day limits, and eat it after you kill it. The health department considers them rodents and will sometimes issue permits to kill them year round in specific areas, you don't need to eat them. Agriculture law allows you to kill any non-endangered species that is a pest (this is what allows you to kill mice as well).

Remember the above is MN law as of 20 years ago when I last checked. Such law changes from time to time and place to place.

Really depends state to state or even property by property. For example, you need a permit to hunt squirrels in South Carolina, however you don't need one to hunt coyotes. The reason is 1: your more likely to go after a squirrel than you would a coyote, and 2: coyotes are actually a problem for wildlife in SC and you are actually allowed to kill one on cite. I guess the factors come down to likelihood of being a target and whether or not it's a nuisance.