NYC allows PR's to become police officers. This is mainly to recruit creole and spanish speaking officers.
The military allows you to enlist as PR. For an officers commission you need to be a citizen (you must have taken the oath of allegiance, not beholden to any 'foreign prince'.)
If you enlist in the military, you are eligible for pretty much immediate citizenship (executive order of GWB in times of war). Given the current conflict, a lot of the guys get their citizenship posthume making it easier for their families to receive the pension.
FBI and CIA can get you a GC if you are a good informant regarding the traficking of drugs and weapons. I don't think they can help with citizenship.