"plainclothes detective," 1906, from the rubber-soled shoes they wore (allowing stealthy movement), which were so called from 1863 (gums "rubber shoes" is attested by 1859); from gum (n.1) + shoe (n.).
gumbo
gum-drop
gummy
gump
gumption
gumshoe
gun
gun moll
gunboat
gunfight
gung ho