The Space Museum:

10ep 1 – Shortly after leaving the TARDIS, as the Doctor bends over to examine the ground, you can see Ian's shadow being cast onto the "landscape" behind, plainly showing that it's just a painted backdrop after all.
[Well, what did you expect?]

20ep 1 – William Hartnell takes three tries to say the word "fluorescent".

30ep 1 – Just after the travellers have discovered they can't touch any of the museum exhibits, they see some of the rebels and try to hide from them behind a dummy wearing some kind of pressure suit — which gets noticeably jostled as they move around it.

40ep 1 – The "intangible" TARDIS found in the museum has the St. John Ambulance badge prominent on the right-hand door, whereas the real TARDIS is missing the badge throughout this story.

50ep 2 – The whole basis of the story is undermined when Ian unravels Barbara's cardigan to make a trail through the museum. Since the version of Barbara in the display case was wearing that cardigan, doesn't this mean that they have already successfully changed the future, and could now leave Xeros?
[And thereby sparing us two very dull episodes...!]

60ep 3 – The room where Barbara hides from the Moroks has an extremely flimsy doorframe. Every time the door is closed, or a character so much as touches the frame, it moves or distorts like a piece of thin cardboard (which it probably is, let's be honest).

70ep 4 – In the final scene in the museum, the inside of the police box shell is visible a couple of times (especially when Ian and Barbara emerge to ask the Doctor about the Time/Space Visualiser), revealing that the TARDIS has no back to it!

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