Usage patterns
髭を生やすto grow a beard/moustache
He recently started growing a beard.
髭を剃るto shave one's beard/moustache
I shave my beard every morning before going to work.
髭を蓄えるto keep/wear a beard (formal)
The actor wears an impressive beard.
髭が伸びるfor a beard/moustache to grow long
I hadn't shaved for a while, so my beard grew long.
Expressions
お髭の塵を取るflatter a superior; curry favor with a higher-up