Plane Death - S1-E14
Deliberate mistake: When the bartender shoots the shotgun at Landon, the windows blow out the wrong way. This was likely done for the actors' safety. Also note how Landon flinches a bit off beat from the explosion.
Revealing mistake: In the song "Living Island", we see Judy frog jumping over Pufnstuf. When she goes back down, the wires holding her up and putting her down are visible.
Gortimer vs. The Mobile of Misfortune - S1-E3
Visible crew/equipment: When Stanley is riding the kart in the race, the shadow of the camera equipment is briefly visible bottom right, just before the kart starts to fall apart.
Factual error: Many elements of Marlott's trial and execution are incorrect. The judge sits with two assessors (a European practice, not a British one). He is addressed as 'Your Honour' instead of the correct 'My Lord'. He instructs the jury to find Marlott guilty and then goes straight to sentencing, which would never happen; a judge can instruct a jury to acquit, but not convict. He puts the black cap on himself, instead of an attendant doing it. The form of words for the death sentence is incorrect. The chaplain (who appears to be Catholic, not Anglican) gives a speech at the scaffold.