If that sounds like a silly question, well, it is. But remember that bullets bounce off this man's eyeballs. Is his hair somehow weaker than that? Does he need a kryptonite razor? We're not trying to get too into the biology of Kryptonians, because that's what our fanfiction.net account is for, but we know he has testosterone. He can have a kid, and he gets superboners (see: fanfiction). But Superman never seems to have facial hair. The few times we do see a bearded Superman, like in Man Of Steel, he's clean-shaven 15 seconds later. But how? And no, Gillette's "Soft enough for a woman of Kleenex, strong enough for a Man of Steel" ads aren't canon.
Before you learn the answer, please try to imagine all of the ways Superman could shave. Is one of his superpowers hair follicle control? Does he just rip his hair out while screaming in agony every morning? Is Batman involved somehow?
Does he casually endanger the lives of his next-door neighbors if his bathroom mirror ever develops a dull spot?