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.