For the long-tailed rodent known as a pest, pet, or company mascot, My ASL Dictionary has a cartoon of a famous Grey-haired Artist and Entrepreneur in a Florida Jacket showing the ASL hand sign for MOUSE, RAT, and RODENT. Index finger flicks across the nose twice, mimicking the twitching of a mouse’s nose.
Another variation hand sign for RAT is to flick the R hand across the nose.