Continuity mistake: When Kaylee asks Inara whether or not the Jiangyin Prairie Paradise plate would make a nice gift for Simon, the position of the plate rotates approximately 45 degrees to the right between shots in Kaylee's hands. (00:07:55)
Suggested correction: I just watched this episode, and the camera was off Kaylee long enough to rotate the plate in her hand, especially if she's fidgeting with it in her hands.
Continuity mistake: After River brings Simon the hodgeberries, the hair on her left shoulder constantly moves between shots.
Continuity mistake: In the opening scene where Simon is talking to his father about his dedicated source box, a newspaper on the table in front of his father keeps moving between shots.
Continuity mistake: As the crew leave Simon and River behind on Jiangyin, Simon's fringe changes between shots. In some there are 3 curled strands of hair, in others there are 2.
Continuity mistake: When Serenity disembarks from the Alliance ship after getting Shepherd Book stitched up, the lights on the port (left) and starboard (right) sides of the ship are reversed - the port lights are green when they should be red and the starboard lights are red when they should be green.
Revealing mistake: In the scene where Simon's dad bails him out of prison, there is one shot showing a very long corridor. It is only seen in this one shot, doesn't appear to have any depth to it and there are no people walking in it. That's pretty unlikely for an Alliance prison. I'm thinking that it is a backdrop.
