(at around 52 mins) When Dr. Selvig is in the mental hospital and doing his presentation on the Convergence, there are several misspellings on the board (for instance his spelling of "Joutenheim" actually begins with a "Y.") This is possibly deliberate due to his delusional state.
(at around 25 mins) When Jane returns and Darcy has called the police, Jane scolds her with "You call the police, they call the Feds!". Britain is not a Federated system and thus has not Federal Law enforcement.
(at around 1h 22 mins) When Jane Foster laments about how she wishes she never found the Aether, Thor claims that Malekith would have found it sooner. This is incorrect because Malekith, along with all the Dark Elves, are clearly seen being reawakened from cryo-stasis after Jane first interacts with the Aether (at around 22 mins).
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.
(at around 46 mins) 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.
(at around 1h 17 mins) When Loki is nearly dragged into the portal after pushing Jane backwards to protect her his hands are empty. When Thor then knocks him out of harms way he lands with his dagger in his right hand.
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.
(at around 1h 4 mins) 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.
(at around 23 mins) 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.
(at around 21 mins) When Jane is pulled into the darkness and first sees and approaches the Aether, she is carrying her device in her right hand. When she bends down to look at it, the device is in her left hand.
(at around 46 mins) 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.
In the first scenes where Jane encounters the Aether, as she floats in it on her back she is wearing the armor given to her later in the story from Asgard. The armor is on the right side of her body instead of her regular Earth dress.
(at around 1h 30 mins) Thor is shown boarding the London Underground at Charing Cross station, asking a commuter how to get to Greenwich. She says he can "take this train three stops," but that's not possible. Two lines pass through Charring Cross. One is the Bakerloo Line, where three stops one way would arrive at Regent's Park, and three stops the other way is Lambeth North, and the Northern Line, where three stops one way is Goodge Street, and three stops the other way is Kennington, none of which are near Greenwich. The nearest Underground station to Greenwich is actually North Greenwich, which is on the Jubilee Line. To get there from Charring Cross, Thor would have needed to make at least one change of train after leaving Charring Cross, and travel a minimum of eight stops. No London Underground trains go directly from Charring Cross to Greenwich.
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.
(at around 1h 7 mins) 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.
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.
(at around 49 mins) 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.
(at around 1h 40 mins) 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.