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.
Continuity mistake: When Jimmy is in his lab, in the shot where the camera zooms into his eye, in the shot just before that shot, notice the digital clock on Jimmy's counter. The seconds part jumps from the 30's to the 40's twice in two different shots.
Continuity mistake: When Dana and Johnny are drinking wine, the amount in the glasses change between shots. Also whenever they pretend to drink, if you watch closely the liquid in the glasses doesn't even so much as touch their lips when they drink.
Forever Red - S1-E34
Continuity mistake: Serpentera is different from the Serpentera shown in Series 2, 3, and 4.