Continuity mistake: Lister and Rimmer's bunk room looks exactly the same in the flashback scenes during this series as it does in the present time. In the first series it looked completely different, so surely the look should have changed slightly for the flashback scenes?
Audio problem: During the conversation in which Holly wants to prove he has an IQ of 6000 (while Lister and the Cat are scrubbing the floor), there is a brief moment where you can see the TV Holly is on (at an angle). You can hear sound coming from him, but his lips aren't moving.
Other mistake: The tax letter and Rimmer's mother's letter are both already opened before Lister takes the letters out. As there is no one else aboard Red Dwarf, the letters would not have been checked by anyone so they should still be sealed. Also, not all of the letters are already open, just the two that Rimmer receives.
Revealing mistake: During filming of the this episode, Craig Charles was called away for the birth of his son, Jack, so production manager Mike Agnew stepped in. Although Lister is wearing a space suit at the time you can still tell which part is Mike thanks to a continuity error - Mike forgot to put on Lister's plaster cast.