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1590s, fusse, first attested in fusball "puff ball of tiny spores," of uncertain origin. Meaning "the police" is American English, 1929, underworld slang, origin and connection to the older word unknown. Perhaps a variant of fuss, with a notion of "hard to please."
A police officer; the police: Cops must be annihilated. Kill the Fascist fuzz
[1930+; origin unknown; the form fuzey is found at about the same time]