Green’s Dictionary of Slang

1. a false accusation.

1950

Goldin et al. DAUL 36/2: Bum finger. 1. A false accusation, especially an identification by a victim of a crime that one really did not commit. 2. (P) An unjust accusation of rule violation.

2. an unfair jail sentence.

1992

Bentley & Corbett Prison Sl. 18: Bum Rap When a person is falsely accused or convicted of a crime, it is said he was given a bum rap [...] (Archaic: bum finger).