One infamous example is when they had a Lieutenant commanding a Nebula class starship. Though I think this was explained as the intention was that he was supposed to be a Lt. Commander, the costuming department screwed up on his pips.
To be sure, this junior Lieutenant was never explicitly stated to be commanding the behemoth Prometheus. He just happened to sit in the command chair during a single key scene where the ship and her command structure were in chaos - just like, say, LaForge or Worf might have done during the first season of TNG. One might imagine two or three more or less reasonable scenarios:
1) The ship's captain and the guest star, terraformer Seyetlik were the same person. Seyetlik's demigod personality would make it quite understandable that he would never don a proper uniform, yet otherwise his behavior was quite in line with him being the CO of the ship. In that scene, Seyetlik was obviously unavailable, and it would also obviously have been difficult for the poor Lieutenant to be more assertive because what this dubious Sisko character was asking him to do was act against the will of his CO.
2) The ship's captain was a Starfleet hero. So naturally he or she would be lying unconscious at the shuttlebay door, having failed to stop Seyetlik's suicide flight despite a valiant try.
3) Seyetlik was disliked, to put it mildly. None of the crew wanted anything to do with him - even the ship's doctor refused to attend to his wife's ailments! And the skipper would not honor him with his or her presence no matter what. The poor Lieutenant had to stand in...
As for access to the Omega secret, I guess Starfleet could always err on the side of caution. If the personnel in the vicinity of an Omega incident did not have proper clearance and training, it would probably be better not to let them do anything; their starship might "put them on hold" for the duration of the crisis and summon more competent help from elsewhere. Not an option in the Delta Quadrant, of course.
Timo Saloniemi