When Simon is saving the patient who flatlined in the hospital he turns to take the defibrillator from the nurse, and when he turns back to the patient he has a syringe in his mouth that neither he nor the nurse put there.
The goof item below may give away important plot points.
Contrary to Reynolds' threat, Jayne would not get 'sucked' out of the partially open air lock hatch and thus not get a space age 'keelhauling'. Since the hatch was already cracked open while still in the atmosphere, there would be no explosive decompression (which would be the actual mechanism; 'tossed' out along with the air rushing out, rather than 'sucked out'). Instead, there would have been a gradual decompression while Serenity climbed into ever thinner air, and Jayne would eventually have suffocated/frozen to death inside the air lock.