A Moustache is a piece of facial hair grown on the upper lip. In Operation: S.H.A.V.E., living Mustaches tried to take over the world by implanting themselves on people's faces and brainwashing their victims, making them obsessed with their mustache. They attacked and possessed Sector T in Texas. When confronted at an office building the mustaches had taken over, the leader of the mustaches explains that they only want to have a home on people's faces, having fallen out of style and shaved off by their previous owners. Numbuh 3 took pity on them, and Sector V had all the mustaches taken to the Antarctic and put on penguins.
They're very similar to the DC comics villain, Starro the Conqueror, since both latch on to people's faces to control them.