i Register
In some senses, gumshoe is marked as slang, US. Watch for register when choosing this word.
noun
A sneaker or rubber overshoe.
A detective.
"Who's this gumshoe guy from the bush league tailin' us?"
verb
To act as a detective.
The next thing they did was to send a man sent down there from Baltimore and he spent a good deal of time gumshoeing around in that county trying to get another individual put in as secretary-treasurer.
But these days, more and more women are gumshoeing through the pages of murder mysteries.