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.]
Reseda Rose - S1-E6
Continuity mistake: While Bill and Ralph drive around checking on the FBI's list of addresses, in the closeups the mirror Ralph uses for his visions is attached to the sun visor with metal clips, but each time Ralph pushes the visor back up the metal clips are gone.
Tiptoe Through the Penfolds - S5-E3
Continuity mistake: Although Danger Mouse is supposed to be on his own, when he turns a corner in the car, in the overhead shot, you can see Penfold sat next to him.