The Long Legs of the Law - S2-E1
Revealing mistake: At the very beginning of the episode, Leonard Pearce is waiting for the cue for his character to start looking for his teeth. He's standing in the living room, between his room and Del and Rodney's room. (00:00:25)
Revealing mistake: At the point where someone gets a small bruise on the upper arm, the crew are all in a pile and thrashing about on the floor. There are 5 pairs of legs instead of 4. Arthur is in his pajamas, Zaphod is in orange trousers, Trillion is lying on top of Ford and Ford is too far away for them to be his legs. [The fifth pair of legs belong to the person playing zaphod's third arm - doesn't mean it's not a mistake, but that's where they come from.]
Heaven is in Your Genes - S3-E6
Revealing mistake: While Ralph is in mid-air, en route to the airport cafe, something dips into view at the top left corner of the screen. Then when Ralph takes off from that cafe, two shots later as he's in mid-air, we can see the shadow of the helicopter filming the background stock footage, but it's been flipped so the helicopter is flying backwards, and when more background footage is shown again the helicopter shadow is flying properly. (00:13:15)