Continuity mistake: The time that the events involving the crew of Red Dwarf keeps changing throughout the series. At various times, references are made to the date, which is stated as being in the 21st, 22nd or 23rd centuries. Examples: In 'Stasis Leak', the date reference is stated as being 2077 (21st century). But in 'Ouroboros', the date is stated as being 26th November 2155 (22nd century). Also in 'DNA', Lister says he is an enlightened 23rd century guy.
Deliberate mistake: In the first few series, it is suggested Lister and Kochanski barely know each other. In 'Balance of Power' Holly says that Lister has only ever spoken 173 words to her. However, by series 4 Kochanski is referred to as Lister's ex-girlfriend with whom he shared a relationship. In the series 4 documentary, Doug Naylor explains that Lister's attitude towards Kochanski was immature in the first series which is why they changed his back story from wanting to be with Kochanski to actually having been with her by series 4.
Continuity mistake: When Rimmer and Lister leave their sleeping quarters, there are two pictures on the wall between the door and the window and a canister on the wall beside the beds. When they are next in the room the items are gone, and at The End of the episode they have all returned. (00:09:20 - 00:12:50)
Continuity mistake: At the start of the 'Welcome Back George McIntyre' reception scene, Lister, Chen and Selby are throwing things at Rimmer, then Rimmer looks around. In the next shot, Lister and Chen are looking round to their left, Selby is stabbing between his fingers, Rimmer is now looking straight forwards and Petersen has appeared from nowhere. (00:09:40)
Continuity mistake: When the Captain is making his speech for George McIntyre, he takes his hand out of his pocket and touches his glass. In the next shot, his hand is back in his pocket again. (00:10:35)
Continuity mistake: When George McIntye finishes his speech, he is standing. When the camera cuts back, he is sitting down. (00:11:55)
Visible crew/equipment: When Todhunter asks everyone to stand for the cutting of the cake, a red boom mic can be seen in the top left corner of the screen. (00:12:05)

Continuity mistake: In George's memorial service, the extras are totally different in the shots from George's point of view to the shots from the back of the room. Look for the woman with brunette hair and orange overalls - she's there in the forward shots but missing in the reverses. The chairs, tables, props, costumes and character placements are all completely discontinuous. (00:12:15)
Continuity mistake: When Rimmer stresses out while preparing for his exam, he puts his hands on his face, and screams whilst starting to bend over. In the next shot, Lister wakes up, and Rimmer is suddenly standing upright. (00:13:20)

Continuity mistake: When Rimmer leaves for his exam, the table in the room only has an ashtray on it. Moments later, when Lister locks the door, a magazine, a can and a mug have appeared on the table. (00:14:30)
Suggested correction: Actually at 13:52, you can see the mug and magazines on the table. The can is just too far off screen to the left to see.