Superman can fly, shoot eyes with lasers, bend steel, see through almost everything, invert time and keep getting movies, even if most of them are terrible. But can he shave?
If that sounds like a silly question, well, it is. But remember that bullets are bouncing on this man’s eyes. Is his hair weaker than that? Does he need a kryptonite razor? We do not try to get into Kryptonian biology too much because that’s our fanfiction.net account, but we know it has testosterone. He can have a child and he gets super-lights (see: fanfiction). But Superman never seems to have facial hair.
In the golden age of comics, Superman’s hair did not grow on Earth. There was even a whole issue dedicated to this at one point – Clark Kent was followed by a journalist / photographer for a rival company. She was one of the sleaze artists in the company – still gossiping.
Her hired investigator continued to take pictures of Clark, and she measured her hair in each of them. She discovered that her hair had never grown a centimeter and came to the obvious conclusion: it was a wig. She then took a picture of him in his apartment – completely bald. She tried to “take him out” on national television by taking off his wig and found happiness when it was revealed that it was his hair. He bit his nose by telling all the viewers that he would explain everything … that night, in the news of Planet magazine (I think at the time, he worked for the television section of the Daily Planet – it was the golden age, and they had one).
On the news, he revealed that he was wearing a white ball wig to prepare Lex Luthor for a police attack or something like that. The problem ended when the journalist swore to understand why his hair never grew.
I think it disappeared in the 80s or 90s, and has yet to come back. At one point in the gun, however, Superman could no longer grow hair, and a whole story (which seemed likely to produce a new lame oversight) was based around that.