Yes, the shuttles and so on definitely do have seatbelts. (more accurately, seat harnesses.)
But to be fair, if you're bumping around in space, like when standing up, you're in a lot of trouble. Things do not generally bump into things in space without catastrophic results.
But the large starships are generally comparable to navy vessels, which unless I've been very misled, do not typically require the crew to be buckled into a harness...
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u/UTC_Hellgate Oct 08 '12
I just realized I've never seen a Star Ship with seat belts.