We could probably reason that Superman's hair isn't quite as invulnerable as he is. It's really strong, sure, hair is mostly "dead" it's essentially a "waste product" of our bodies and contains no living cells (the only living cells are in the follicle) so while it may be strong and may have some level of resistance to damage from fire, explosions, etc. it may still be possible to cut it with a strong enough sharp-edged blade of some sort or with powerful lasers. The comic once suggested he reflected his heat vision off mirrors to "cut" his hair (and shave) but this strikes me as needlessly complicated and, well, silly, that as smart and capable as Superman is he's able to precisely aim his heat vision in a manner to shave his face without burning his face off or give himself a nice, clean and even, haircut.
In an episode of "Smallville" a possessed Lana tries to steal some of Clark's hair (for what purpose I don't exactly recall) and she breaks a pair of scissors in her attempt to do so, Clark offers some hair by pulling it out of his scalp covering for the scissors by dismissing them as old or broken already.
There's also a "possibility" that the spaceship Clark was sent in contained Kryptonian materials that could be made into shaving/cutting devices or just outright ARE those things made from Kryptonian materials and thus capable of harming Kal-El for the same reasons he's impacted by other Kryptonian artifacts and remains. Small or trace amounts of kryptonite can shave him/cut his hair without causing him any serious harm from the exposure to kryptonite or there could just be, yet another, form of kryptonite that can cut him/his hair bur otherwise has no other effects. (Hey, if there's a version of kryptonite that causes some random effect on him, or can turn him gay, then there's got to be one that can cut his hair. Pick a color and that's the one that does it. I'll go with chartreuse because I like that word.)
