Continuity mistake: When the cast are at the Miranda colony and discover that everyone is dead, there is a rotating shot of River as she realises what happened. Prior to this there is a brief closeup shot of the rest of the cast as they turn to face her. Jayne turns to his right, but in the following rotating shot of River he has turned from his left.
Continuity mistake: When the crew are in the cockpit talking about Miranda, Jayne is in the background holding his head with his left hand. 5-6 shots later he is holding it with his right hand. On the cast commentary of the DVD, Adam Baldwin says that he did this on purpose "just to mess with continuity".
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.
Continuity mistake: While Zoe determines the location of Miranda, there are two shots of Jayne with his arms down by his sides - one where he is standing behind Inara and one with Wash in the foreground. When it cuts back to the shot of him standing behind Inara, his left hand is on his belt. His hand then returns to its original place when it cuts back to Wash.
Continuity mistake: During the last fight with the Reavers, Inara is holding a blaster gun shaped like a bow and arrow in close up shots. In any shot that shows the whole crew, the blaster changes into an actual bow and arrow. The producers didn't like the bow so they actually enhanced it with CGI to make it bigger and "cooler", so those shots of the whole crew are probably shots where they forgot to add the CGI or assumed people wouldn't notice.
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.)
Continuity mistake: Seeing as how the original sets for the ship's interior from "Firefly" were demolished after the show got cancelled, the interior had to be re-built from scratch for Serenity. Because of this, there are some noticeable differences between the two. For example: The long corridor that holds the entrances to the crew's sleeping quarters is a lot wider. The hand rail on the stairs leading from the infirmary to the cargo bay is missing. The back wall of the cargo bay, the entire engine room (including Kaylee's hammock), some of the table settings in the kitchen and the furniture in the lounge area are all different to how they were in "Firefly".
Factual error: At the end of River's first fight scene, Simon says a phrase that renders River unconscious. The phrase is "Eta kuram na smekh" and means "This is laughter for chickens", or "This is absurd" in Russian according to Joss Whedon. However, the subtitles on the DVD for the phrase are "(SPEAKING CHINESE)", not "(SPEAKING RUSSIAN)" like they should be. In fact, shouldn't the subtitles just say "Eta kuram na smekh"?
Continuity mistake: Near the beginning, when the assassin hits the researcher in the back and then slams the sword in front of him, the blade has very little blood on it. Just before the scientist falls on it, we see it and there is blood thickly coated over most of the top of it, presumably from the two men he'd slaughtered moments before.
Visible crew/equipment: In the vault scene, there's a part when the camera shifts and you can see the area outside the cylindrical vault set in the upper left of the screen.
Continuity mistake: When the crew are sitting in the dining area of the ship after discovering the recording on Miranda, Jayne slides the bottle he's drinking from to Simon. The bottle spins as it slides across the table, and when Simon grabs it the label can be seen between his fingers. The shot cuts away, and when it cuts back, Simon is suddenly holding it with the label away from the camera.