Green’s Dictionary of Slang

a moustache.

1943		J. O’Hara ‘Now We Know’ in New Yorker 5 June 19/2: [A] rather sad-eyed Jew with what Mary called a little muzzy that made him look somewhat like an ugly Ronald Colman.
1975		A. Goldman Lenny Bruce 13: All spic broads, guinea broads, Jew broads [...] got that little mussy.
1998		K. Sampson Awaydays 67: How could anyone ever think a muzzy looked cool?
2001		K. Sampson Outlaws (ms.) 92: He used to look like Bob Carolgees, years ago when he had a bad muzzy and a sheepie.