Revealing mistake: Toward the beginning of the episode, when we first see Inara at the controls of her shuttle, she is holding the steering wheel upside down as she flies away.
Bushwhacked - S1-E2
Revealing mistake: As Mal rolls the survivor of the derelict transport onto the floor, the survivor moves his left hand onto his chest when he is supposed to be unconscious.
War Stories - S1-E9
Revealing mistake: During the firefight on Niska's ship, when Simon sees a guard coming and says to Sheppherd Book, "Book, we, uh-", a dead guard on the floor moves his legs as the guard that Simon saw fires his weapon. (00:37:10)
Revealing mistake: When Nandi confronts Rance stealing the baby, it is a plastic doll and not a live child in his arms. (00:35:40)
Revealing mistake: In the scene where Mal and Zoë exit the train, look closely at the woman in the yellow veil to the left of the screen holding a baby - she walks into a passing vehicle knocking her baby on its head (revealing it is really a doll).
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.
Revealing mistake: When the camera follows a man through a curtain to reveal the Eavesdown Docks on Persephone, an extra in a white shirt, black vest and straw hat can be seen standing still on the left looking straight into the camera. He then looks away and starts walking. He was waiting for his cue from the director. Several seconds later, the same extra is seen again behind Mal and Kaylee walking in the opposite direction he was just going, giving the impression that the scene wasn't filmed on a location but in fact a small set and they could only afford so many extras.
Revealing mistake: When Simon prepares an injection for Jayne, you can see him insert it and draw back a considerable amount of fluid from the medicine vial, but the level of liquid in the vial does not drop.
Revealing mistake: In a travelling shot, you can see an extra in the middle of the street, looking at the camera and waiting for his cue to start walking. (00:17:26)
Chosen answer: One of the premises of the show is that the two major Earth societies that survived, and ultimately merged, were the Chinese and Americans. As a result the common language includes much Chinese. Clothing, signage, and building designs also have a heavy Chinese influence.