The Friendly Physician - S2-E29
Plot hole: The scientist yells for Igor to capture the escaping Castaways, to which Igor responds "capture them yourself" in Ginger's voice. The scientist appears surprised to find out Igor is not Igor, but how can this be? He had to be the one to pull the lever to switch everyone's personalities and bodies around, so he should have already known.
Plot hole: In order to build up suspense, Virgil and Brains stand on the Crab Logger as its dangerous fuel is pumped out, despite the fact that it is hanging on the edge of a precipice. In fact they could have left as soon as they had plumbed in the pipes to empty the fuel tanks - the whole procedure is controlled by Scott in Thunderbird 1 and they make absolutely no contribution to the operation at all. They do not need to disconnect the pipes; they are left behind and dropped from Thunderbird 1 as planned. They just stand there chatting about how much danger they are in until jumping clear at the very last second. Silly buggers.
Visible crew/equipment: In the first scene of the episode, a crew member is visible through an open door beside the staircase.
Double, Double, Toil and Trouble - S4-E4
Revealing mistake: As Serena gets hit by the pie her real blonde hair shows underneath, also, she wears minimal make-up for the only time.
The Addams Family Goes to School - S1-E1
Continuity mistake: After Mr. Hilliard goes through gate and nears the house, it's really at the real house. A second later, he's approaching the now-familiar front porch on the studio set. This is easy to tell because the layout of the two porches are radically different.
Emergency Marineville - S1-E11
Continuity mistake: When the second group of Spearhead jets take off from Marineville there are only two jets, but in the next shot you can see three jets, and in the next shot there are two again.
Flipper and the Bounty - S2-E13
Visible crew/equipment: After Bud wakes up all alone in Bligh's cabin, he soon realises all the doors are locked. Then, just as Bud rushes over to the stern window (particularly the second time), a seated crew member is visible at the bottom right corner of the screen; we can see the movement of a leg and two hands (so, it's not a mannequin).