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.
When Malekith enters the Asgard throne room, he detonates a grenade which destroys Odin's throne. However, when Odin and a host of soldiers enters the throne room seconds later, the throne appears to be still intact.
When Kurse throws a huge boulder at Thor in their final battle on the Dark World, Thor sustains a scar on his left forehead. During the rest of the movie, the scar moves to his right forehead and changes shapes. Most noticeable is when Thor is holding Loki in a death scene, the scar is half the length of his eye, rest of the movie the scar is the length of his eye.
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.
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.
After the Aether enters Jane she returns to London, where its power protects her from the pouring rain - and yet an overhead view shows that she and Darcy are standing well away from the center of the circular dry zone.
In battles throughout the film, multiple stone columns are destroyed without any collapse of the building above. Perhaps in Asgardian architecture these columns are merely decorative, but their support is definitely required at Greenwich Palace.
During Fandral's "For Asgard" fight scene, there are two consecutive green-screened shots of him landing between soldiers on the flying craft, one with a soldier on the left of the frame, the other with a soldier on the right. They clearly use the same footage, flopped to make them look like different angles, with slightly different backgrounds keyed in. Fandral flies in from the wrong angle in the second shot, is facing the other way, his sword jumps to the other side of his body, and it's clearly the same guard in both shots.
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. 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.