Green’s Dictionary of Slang

fuzz n.1

also fuz
[? SE fuss, which a police officer makes; or ? man with the fuzzy balls under balls n.; note without detailed context senses 1 and 2 can be indistinguishable]

1. [1920s+] (orig. US) a police officer; the police in general.

2. [1930s–40s] (US) a detective.

In derivatives

fuzzmobile (n.) [-mobile sfx]

[1970s–80s] (US) a police car.