On Miranda, around the time of River's "Make me a stone" speech, Jayne turns around to the left. In the next shot, he turns around to the right. In the commentary for the special edition, he admits that he did it on purpose.
At the start of Dr. Caron's holographic recording, there is a noticeable delay before she says, "... .and you can see, it isn't what we thought". When the recording is replayed and broadcast at the end, the delay is gone.
After they do the 'Barn Swallow' technique after being chased by the Reavers in the beginning, Jayne is in the Mule. When Kaylee comes down the stairs, Jayne is also on the stairs. After Simon hits Mal, it goes back to Jayne pouring alcohol on his wound, and he is back in the Mule.
In the wide shot of the bar scene that Mr. Universe is watching, the camera operator is visible, in an orange tee shirt, in the upper left hand corner of the shot just before he says the line, "And she falls asleep."
On Mr. Universe's planet the audience is told Kaylee rigged the blast doors so no one could open them once they are sealed. Yet, right after Mal asks about River, the blast doors do open. The doors were obviously malfunctioning earlier when they only closed halfway.
When Mal and Zoe talk about the man left behind during the bank job, an orange flashing light reflects off Mal's face, but not off Zoe's, even though they're standing opposite each other. This light is fire from the bridge windows as Serenity leaves the planet. This is explained in the commentary.
When Serenity is hit by the Reavers' EMP and begins to crash onto Mr. Universe's planet, Wash strains against the force of rotation while he reaches for a panel of buttons on his console. When his hand hits the panel, the entire panel moves; it was made of cardboard.
When the doors open to reveal River standing over the dead bodies of the Reavers, she is in the middle of the room, holding weapons still dripping with blood. The Alliance forces haven't arrived yet and no one from Serenity opened the blast doors, yet they open by themselves.
In Firefly: Serenity (2002), Simon explained to the crew that "some men" told him that they could sneak River out of the lab in cryo where he could rendezvous with her on Persephone. He met up with them later, or else there would be no reason for her to still be in cryo when they meet her.
In the big gun battle between the crew and the Reavers, just as Zoe fires her shotgun for the first time, Inara is in the background, holding a bow and arrow instead of the bolt thrower she is supposed to be using. The bow-and-arrow was the original weapon used, but it was replaced in post-production due to test audience reactions.