Ok...
Superman cuts his hair with his Heat Vision. That's the answer, simple as that.
Ok ok, but what if I wanted to cut his hair?
Well that gets into the question of Superman's powers.
First off, there is the simple answer of "Kryptonite", which really shouldn't work, but I don't write superman and DC isn't consistent in how his powers actually work so in general Kryptonite will weaken him and you can then cut his hair...How, that works who knows, since hair is dead and can't degenerate any further which is the explanation of how Kryptonite works...
Secondly there are 2 modes for his invulnerability. The first is he is just Super dense and therefor his hair is also super dense, but still somehow blows in the wind like regular hair. The second is that he generates a field around him that makes those things within a certain distance from his body, also invulnerable, similar to how Ki in Dragon Ball works.
In the second mode, you'd have to have him redirect it if he can, use some sort of science to pull it away from his head, or just drain him down to the point where he can no longer output the field. If his hair retains a part of this power then him pulling it out and giving it to someone, the result would be after however long, eventually that piece of hair would just stop producing the field since it is no longer getting energy and then you could cut it as would be able to cut anything.
In the first mode... there are two possible ways to tackle this problem. The first is, just get a really sharp blade. You cut things by way of wedges that have an edge so thing that it can get between the atoms of the thing it is trying to cut and split apart. A blade sharpened enough should theoretically be able to cut anything, no matter how dense it is...
The other way is to decrease the density, or rather make the space between the atoms bigger than they were before and then a thicker edge can cut his hair. How does one go about making something less dense? Heat it up. The question then is which point will he come to first? Will his hair burn away first or will it be less dense enough to cut? Given how dense superman supposedly is (while not weighing more mind you ^.^) I think the burn away first is the more likely answer...
How hot it needs to be I don't think there is any data with regards to that directly, but consider superman can withstand the heat of the sun without his hair burning away. So you need to create a heat hotter than that and you need to focus it into something that can be controlled enough to cut hair rather than just shaving it all off. The closest thing to being able to do that would be the plasma sword concept idea that some people have come up with to mimic a light saber that to pull off would be the size of a sky scraper...
So how does a "normal blade" cut his hair? It doesn't.
Unless his invulnerability is all that field in which case it's simply a matter of getting past that field, which Superman could possibly control at will...