(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.
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.
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 7 mins) Sif stops a spear fired at Thor with her shield. The spear goes through it. When she is gathering the prisoners later, the shield is intact with no hole in it.
(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.
Mainstream news broadcasts in the UK would never show nudity, even pixelated. Any live broadcast would pan away from a streaker immediately and not cut back until police had caught him and covered him with a jacket.
At about 1h 14m the Aether is extracted from Jane and the camera zooms into her eyes and an image of the Earth and Moon appears. The relative size and distance is very much at odds with reality. In fact the moon is surprisingly distant from the Earth and any image with the Earth in a full shot like that would not have the moon in the same frame unless the moon was very tiny and on the far side of the Earth.
No one is able to touch Jane while the Aether is still within her and yet Thor and Jane are able to kiss at 37:30.
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.
In the official trailer, the shot featuring the accident involving a yellow SAAB 900 convertible is a mirrored view from the scene in the movie.
When Queen Frigga dies protecting Jane, as Odin is holding Frigga, you can see her arm rising & lowering from her breathing while she's laying there dead.
The Eurofighter pilot reports 'Missile away', having fired on the dark ship 'moored' on Greenwich embankment, but he pronounces the word in American style: "missal"-he's a British RAF pilot, and would therefore pronounce it as written: "missile".
(at 1h 05m) When Thor says "Out of the two of us, which one can actually fly?" The part "can actually fly" doesn't get mouthed by Thor in the scene.
(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 Charing 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 stations to Greenwich are either Greenwich or Cutty Sark on the Docklands Light Rail. To get there from Charing Cross, Thor would have to go one stop south on either line to Embankment, then take the Circle or District line five stops east to Monument, then change for the DLR Lewisham Branch and take that to Greenwich.
In their escape, Thor didn't tell Loki that they were going to abandon the big ship and jump to a smaller ship. This small ship barely fit into Loki's secret pathway. If Loki thought they were going to continue on in the big ship, he would have known that there was no way they could escape through the secret pathway and never would have gone on with the plan.
Despite being labelled by Odin as one of the most powerful artifacts in the universe, the Aether is seemingly unable to do any form of damage against Thor during the final fight against Malekith.
The Bifrost bridge is repaired without any explanation, despite being broken in the previous films. Making confusing why Thor didn't traveled back to Earth, as he had promised in the original movie.
In the post-credit scene, it is revealed that the red Infinity Stone has been sent to the Collector. However that doesn't make sense, because at the time, Loki was impersonating the king of Asgard Odin (taking his place at the throne of Asgard without anyone knowing). Since Loki is a selfish character, he would had wanted such thing for himself, to further his own power. Given how on all movies he always tries to keep the Cosmic Cube with the Infinity Stone to himself. It doesn't make sense for Loki to give a powerful artifact like that away.
(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).