For some reason, information concerning lockouts on hard mode flashpoints seem to be hard to come about, so this is as much of a reference for me as it is for anyone else.
Normal flashpoints have no lockout timers.
Hard mode flashpoints can be done once per day. A new day starts, and the lockouts reset at 7am EST / 4am PST / 8pm Singapore time / 11pm Melbourne time each day.
Operations lockout timings can be view in your log. 8 and 16 man operations do not share lockout timers.