A gumshoe is a detective — usually a private one, the kind who walks the streets, asks questions, and watches people, not a police officer in uniform. The word is old American slang, a little playful and a little old-fashioned; you meet it in crime stories and films more than in everyday talk. People called detectives gumshoes because they wore quiet shoes with soft rubber soles (the “gum” is that sticky rubber on the bottom), so they could follow someone without making a noise. Though actually, nobody at a real police station would put gumshoe on a business card; it is the word from the story, not from the office. You can also use it as a verb: to gumshoe is to go around investigating, like a detective. Informal, and slightly jokey.

私家偵探（美式口語、偏老派；不是制服警察）