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.
Out of Gas - S1-E5
Continuity mistake: In the flashback where Mal and Zoë meet Jayne for the first time, every close-up of Mal and the one close-up of Zoë shows their hair blowing in the wind. But in the shots of them together, their hair is completely still.
Jaynestown - S1-E4
Visible crew/equipment: When Simon poses as a buyer and gets a muddy hand print on the back of his suit, he and the gang are given a tour of the mud complex. In the first shot, the shadow of a camera can be seen on the ground in the very bottom left corner of the screen.
Continuity mistake: When Mal and Nandi are talking and drinking on the couch, the position of Mal's left arm keeps changing between shots. In both the closeup shots of Mal and Nandi, his arm is resting next to her on the couch, but in the shot of them both, his arm is behind her.
War Stories - S1-E9
Continuity mistake: As Zoë is walking through the halls of Niska's ship to give him the money, her hands keep switching from raised to down by her sides.
The Message - S1-E15
Visible crew/equipment: Tracey threatens the crew with a gun on the bridge when he hears that they're about to call the cops. He says that he'll blow a hole right through Wash if they do. There is a shot of Mal saying "You mailed your ugly business to Zoë and me, Tracey." As he says this, a crew member's hand can be seen above them (through Serenity's windshield on the left of the screen) grabbing one of the pieces of fallen ice.
Continuity mistake: In the scene where Jayne is telling Simon and River to keep clear of Saffron, he is holding some things wrapped in tin foil (possibly food). He places them on a table next to him. River then calls Saffron a liar. In the following shot of Jayne, one of the food packets has been torn open. River later says that "Jayne" is a girl's name. The food packet that was torn open is now intact.
War Stories - S1-E9
Continuity mistake: During the torture scene, Wash has quite a large stomach. But after he is saved by Zoë, his stomach is considerably smaller.
Continuity mistake: When Shepherd Book tells everyone about the electrical nets, Jayne says, "One day you're gonna tell us all how a preacher knows so damn much about crime." Shepherd then turns to leave. But in the immediate close up of him and Mal, he turns to leave again.
Continuity mistake: When Early drops an unconscious Mal down the hatch to his room, in the shot from above Mal falls with his arms by his side, but in the next shot when he lands, his arms are above his head and fall to his sides when he hits the ground.
Continuity mistake: After River brings Simon the hodgeberries, the hair on her left shoulder constantly moves between shots.
Continuity mistake: In the opening scene where Simon is talking to his father about his dedicated source box, a newspaper on the table in front of his father keeps moving between shots.
Continuity mistake: As the crew leave Simon and River behind on Jiangyin, Simon's fringe changes between shots. In some there are 3 curled strands of hair, in others there are 2.
Visible crew/equipment: When the fake ambulance that the crew made to infiltrate St. Mary's hospital lands at the main entrance, it lands on flat foot pads. But in the following shots the casters that were used to move the prop around the set are visible beneath the landing feet.
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.
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.