Deliberate mistake: After the Crazy Ivan, where they escape from the Reaver ship, the shot of Mal, Wash and Zoe celebrating on the bridge. Wash has his hands out in front of him like he's still holding the stick, but there's nothing in his hands. In the commentary, Joss Whedon mentions that, in order to get the shot right, they had to move the seat back far enough that Alan Tudyk couldn't reach the controls, so they instructed him to mime it.
Visible crew/equipment: When Simon asks Mal why he is so scared of the Alliance, Mal punches him. On the bottom left side of the screen, someone's hands are visible catching Simon as he falls.
Continuity mistake: At the beginning of the episode when Mal, Jayne and Zoë are ransacking the destroyed ship for supplies, after Mal says "Vault's sealed" there is a shot of Mal as he inspects the door hatch before asking Jayne for the explosive gel gun. The ring around the door hatch's handle from the explosive gel gun is already visible in this shot.
Continuity mistake: The first shot we see of Nandi waking up shows her hair loose on the pillow, and her hand curled up on the sheet beside her; in the next shot, a close-up, her hair is mostly tucked behind her ear and her hand is closer to her face than previously. (00:25:10)
Continuity mistake: When Saffron first hits Wash, he falls with his head in the centre of the doorway (from our point of view), but a second later, when she goes to lift his legs through the door so she can shut it, his head is noticeably more to the left (again, from the viewer's point of view). (00:27:05)
Out of Gas - S1-E5
Continuity mistake: One of the candles on Simon's birthday cake keeps blowing out and re-lighting itself between shots.
Jaynestown - S1-E4
Continuity mistake: Right after Simon gives his "box dropping man ape gone wrong" line, as Kaylee says "you're pretty funny", he starts to take a drink. When Kaylee finishes talking, the shot changes to in front of them both and Simon's hand and mug are now in his lap.
Jaynestown - S1-E4
Continuity mistake: When the foreman first slaps Simon on the back, the muddy handprint has quite identifiable finger marks, but in the next shot, when Simon is walking away, the muddy shape has changed. In a subsequent shot, the mudprint has disappeared entirely. (00:04:25 - 00:06:00)
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).
Visible crew/equipment: Right after Kaylee tells Mal that she's not speaking to him there is a wideshot of the ballroom dancers. In the archway at the back of the room Joss Whedon, the director/creator, walks into shot wearing a Hawaiian shirt, shorts and a baseball cap with headphones around his neck. Just as he raises his hands to speak to the dancers the shot cuts to Inara and Atherton.
Continuity mistake: When talking to Saffron after being caught by her husband while holding the Lassiter, Mal refers to her as 'Yo/Saff/Bridge', an amalgam of her three aliases, Yolanda, Saffron, and Bridget. The subtitle reads this as "Speaking Chinese."
The Message - S1-E15
Continuity mistake: At the beginning of the episode, Jayne receives a hat from his mother in the post, and when he takes it out of the package it is covered in wood shavings or some similar packing material. He puts it on his head like that, but for the remainder of the scene, the hat alternates from being covered in shavings to clean and back again.
Continuity mistake: When Nandi is killed, she falls to the floor on her back, her left arm by her side and her right on the floor by her head. We see her in this position a number of times, but when Mal and Inara look at one another at the end of this scene, Nandi's left arm is now lying across her chest. Neither of the other characters did this to her. (00:35:00)
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.
Jaynestown - S1-E4
Continuity mistake: The mud left on Simon's shoulder by the foreman slapping his back disappears soon after, despite him having had no opportunity to change or clean his jacket.
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.
Continuity mistake: After Durran calls the police and Mal and Saffron try to escape, they leave the Bellerophon Estates through a door which uses a crash bar to exit. However, when the guards arrive at the door, it has changed colour and has suddenly started using a regular handle.
War Stories - S1-E9
Continuity mistake: In the scene where Malcolm Reynolds is being tortured, after Zoë takes Wash away, the villain cuts off Malcolm's left ear. The shot changes, and then returns to Malcolm, showing the other side of his face. Now the visible ear, the right one, is missing. The shot changes to the front, and now his left ear is missing again.
The Message - S1-E15
Other mistake: The characters in this show often use the made-up curse words 'gorramn' and 'gorrammit.' Toward the end of this episode, just as Tracey grabs Kaylee as a hostage, he shouts, "Gorrammit, Mal." The subtitles read: "You're a rabbit, Mal." (00:35:50)
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.