Near the start of the film a shot of the London skyline shows the new 'walkie talkie' skyscraper as nearly complete. During the final battle a few shots show it several months earlier, with the building at a much earlier stage in the build. In the final scenes you see the building again, at yet another stage of development.
Jane stands back on Thor's left in the Dark Elves' flying machine. When it flies off and leaves the palace, she completely disappears from view. When she falls to the floor later, she has reappeared in the front of the machine to Thor's right.
When Thor comes to meet Heimdall at his post by the Bifrost, he is wearing a cape flapping around the legs; the figure seen walking towards the building from afar is not wearing a cape and his legs are clearly visible.
After the attack on Asgard by the Dark Elves, Odin enters the Throne Room with about half a dozen guards and Sif. After he blasts off a dark elf, he is seen standing alone; then with 1 guard only. The Lady Sif has disappeared.
On the Asgardian boat on their way to find Malekith, the very weak Jane Foster is seen lying down on her right, then on her left; and her position switches again to the right and the left in between shots.
Thor is shown boarding the London Underground at Charing Cross station, asking a commuter if it is the correct train in order to get to Greenwich. She responds that it is. The nearest station to Greenwich, North Greenwich, is on the Jubilee Line, meaning that Thor would have to make at least one change of train in order to get to his destination. No London Underground trains go directly from Charing Cross to Greenwich.
The goof items below may give away important plot points.
In the first shot of Frigga on the floor when Odin sees her body, her left arm is flung out wider. When Odin cradles her, the arm has moved closer to her shoulder; her right arm and body are also moving as if she's breathing.
When Malekith's vertical black ship moves from the Thames in Greenwich it plows a huge trench along the College's Grand Square, but when the ship is later destroyed the Square is intact with only a black splatter mark where the ship was.
In the last scene "Odin" recounts part of a private conversation he had with Thor at the end of Thor (2011). However, "Odin" is actually being impersonated by Loki, who was not present during the aforementioned conversation and thus could not have known what had been said.