Medieval Europe, hair and facial hair

The obvious one would be the two styles of monastic haircuts. Beyond that, I'm not sure there ever were any *named* styles. You can look at pictures from medieval manuscripts to get some general ideas. I am unaware of variations by profession or social order, even in the late Middle Ages.

However! Just because I don't know about it doesn't mean much. A search on "hair styles in the middle ages" turned up the following
https://www.medievalchronicles.com/medieval-clothing/medieval-hairstyles/
Middle Ages/Renaissance
Middle Ages Hairstyles
Rosalie's Medieval Woman - Hairstyles

None of these cite sources, so use them as inspiration rather than authority. If you're interested, here are some academic works that do cite sources.
What functions and meanings did hair have in the early and high Middle Ages?
https://history.sfsu.edu/sites/default/files/EPF/2010_Peter H. Johnsson.pdf
Celtic Hair History | The Celtic Fringe

So as you see, there's actually quite a bit to be said on the topic!

 

Tonsure is the name, I think, for either style. As for facial hair, I believe one of the references I gave addresses that.

 

I think there's only one pdf. When you say you could not access, do you get a 404 or what?