Oolite Bulletins

I keep getting made an offender for killing certain ships. Which ships should I avoid killing to prevent this?

Well, the obvious ones are Police ships
But seriously attacking any clean ship within the station aegis, or when a Police ship is within scanner range will get you an offender tag. So will taking contraband out of a station.

Edited to add:_ you will need a scanner targeting enhancement to know the legal status of other ships. As a rule though, if he shot you first you'll be OK firing back.