So, Gillette is using the question as part of their marketing campaign - pretty cleverly, I think. They ask famous nerds like Kevin Smith and Bill Nye what their theories are, and they come up with some pretty elaborate (the Mythbusters' Large Hadron Collider Theory) and pretty painful (Kevin Smith's Comet Theory) solutions.
But I have a simpler, more obvious explanation.
Superman gets his powers from the rays of Sol, our yellow G-type sun. Under the rays of a red sun, like that of Krypton, he becomes a normal being, as vulnerable to sharp objects as anybody else. So all he needs to shave (and trim his hair, and wax, for all we know - that is a pretty skintight suit, after all) is to replicate red-sun conditions for a while until his beard softens up, and then go to it. Maybe he carries a scruff until a really good sunset; or he has a special darkroom kitted up with red-sun LEDs.
Of course, that means that for ten minutes a day, Earth is defenseless as Superman grooms himself. Which means that the day you see Superman sporting an elaborate, time-intensive hairstyle is the day he has stopped caring about his job. Get ready to kneel before Zod.
ETA: One of the celebs with a theory is Mayim Bialik, actress and neurobiology PhD. And for those who only know her from the Big Bang Theory...wow. Just wow.