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.
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)
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)
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)
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.