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 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).
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.