I suspect the answer will vary from state to state.
In California, any person convicted of a felony (or of a crime in another jurisdiction that would be a felony in California) is prohibited from employment as a California peace officer [Government Code §1029(a)]. This prohibition holds even if the conviction was sealed, expunged, or set aside. It may also apply to any convictions that were subsequently reduced to a misdemeanor occurring on or after January 1, 2004.

