The Message - S1-E15
Continuity mistake: The cut the doctor makes on Tracey is small, but once Mal is on top of Tracey, the cut is a lot longer.
Continuity mistake: In the close-up shot of the explosive in the sticky substance being applied to the outside of the cargo hold, the pink wire sticks out at the bottom, and in a loop in the middle. In the next shot, from slightly further back, the wire doesn't stick out anywhere, and the shape of the explosive ring has changed and become more oval. (00:05:30)
Out of Gas - S1-E5
Continuity mistake: Kaylee has just put Simon's birthday cake down on the table when the explosion happens, but in many subsequent shots of both the table and the floor in the dining room, the cake is nowhere to be seen.
The Message - S1-E15
Continuity mistake: In the first scene in which the crew is back on Serenity, they are discussing Mal and Zoe's recent "delivery." Jayne is shown taking his hat off. Two shots later Jayne is shown in the background with the hat still on his head. Six shots later Jayne is shown with his hat in his hand again.
War Stories - S1-E9
Continuity mistake: The size of the red marks on Wash's body change from when he first leaves the torture chamber to going into the cockpit of the shuttle.
Continuity mistake: Right at the end of the episode, when River says 'Also - I can kill you with my brain' to Jayne in the infirmary, in the long shots she is leaning much further through the doorway than she is in the close-ups, where her face is barely past the edge. (00:39:40)
Continuity mistake: Just before Serenity disembarks from Triumph near the beginning of the episode, there are some small areas of shrubbery and foliage surrounding the exit ramp. But once Serenity takes off, the shrubbery and foliage is gone.
Bushwhacked - S1-E2
Continuity mistake: At the beginning of the episode, when Inara asks Simon how River is, River's hair is hanging down behind her back on both sides. But when Simon tells Inara that she still won't talk about what was done to her at the Academy, her hair is hanging in front of her right shoulder. A few shots later her hair is hanging behind her back again.
Out of Gas - S1-E5
Continuity mistake: When we see the flashback of Mal and Zoë first entering Serenity's cargo bay, just after Mal has bought the ship, there is a cable on the floor behind them. For a short moment, we see the cable being dragged over the grille right behind them, then the shot changes and there is nothing there at all; the only cable is much further back and lying at a completely different angle. (00:01:55)
Continuity mistake: When Mal is fighting Crow on the ship, he falls over the containers of medical supplies, knocking some of them over. A few shots later when Jayne shoots Crow, the containers are standing upright again.
Continuity mistake: When Inara finds Mal unconscious in his quarters, his head is resting up against the wall. When the angle changes to show her running over to him, he has moved away from the wall and his head is on the floor. After she has called for help, his head is once again up against the wall. (00:31:00)
Factual error: Jayne doesn't need the suit to fire his gun, because ammunition contains an oxidizer that would allow it to be fired even without oxygen present.
Suggested correction: From a different perspective it's not a factual error. Because Jane shot multiple bullets. It means the gun still can be shot after the space suit case got pierced from the first bullet. So it's a possibility that the characters thought wrong (even though it's a bit silly for space travellers to not know this fact but doesn't mean it won't happen).
Visible crew/equipment: In the celebration at the beginning, the shot where Jayne receives the rain stick, a camera crew is briefly visible just to the right of Jayne's head right before he says "I will treasure this."
Continuity mistake: As Mal approaches the desk, saying "we got a couple of DOAs" he puts his left hand directly on the corner. Cut to a shot from behind and his hand's moved about a foot.
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.
Visible crew/equipment: As Wash leave the ship to get supplies, you can see a hand pulling the front of the quad. (00:17:58)
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)
Out of Gas - S1-E5
Continuity mistake: In the first flashback in the episode, Malcolm is showing Zoe Serenity after buying it. You can see on his left arm the stitched-up bullet hole that was from the very first episode when Mal gets shot. This episode would have taken place much later than the flashback.
