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.