Revealing mistake: On Miranda there is a shuttle crashed into the side of a building, but there are no burn marks or a trail from where it had crash landed. The shuttle couldn't possibly have crashed into the building horizontally without crashing diagonally/vertically first, landing horizontally and then sliding into the building. Joss Whedon even states in the director's commentary that some burn marks would have been nice but it only occurred to him about 14 months after it was filmed.
Revealing mistake: When River goes crazy on Serenity, knocks out Jayne and Simon and then holds Mal at gunpoint, you can tell that the gun she is holding is quite heavy because her arm wobbles. But when she cocks the gun it is completely still. In the cast commentary of the DVD, Joss Whedon reveals that the gun was in fact held up by a C-stand for that particular shot because the gun was too heavy for Summer Glau to hold. (Note: This is not the scene with River's choreographed fight in the bar on Miranda.)