Normally I'd offer some choices, but I LIKE pigeons, dangit!
Rather than shoot them, shoot the people who are feeding them...IN THE ASS. When the food source goes away, the pigeons will go in search of more productive meal tickets and everybody's happy except the jackass with an ass full of pellets.
Individually, a pigeon is no messier than any other bird of the same size. The problem is that there are usually a lot of pigeons together, of course. But be assured, it'd be a lot worse if you were dealing with an equal size flock of virtually any species of birds, and especially it'd be bad if you were afflicted with parrots. They're messier than pigeons and a hell of a lot noisier, plus they have a set of bolt cutters on their faces and can give a hell of a bite.
CJ