The scabbard for "Orcrist" is split down the edge side because the tip half of its blade is wider than the lower half towards the hand guard - when Thorin (partially) draws the sword after he finds it in the troll hoard, his fingers are wrapped around the split, putting him in severe risk of cutting himself (you can clearly see the wide part of the blade emerging from the split under his hand). A seasoned warrior like him would not make such a mistake, even if the straight dwarven swords he's used to don't have this feature.
Scenes with Galadriel make a point of showing that she turns in place so that the bottom of her dress wraps beautifully around her legs. When she and Gandalf say goodbye, she turns around and in the process spreads her feet and legs apart, but at the very next shot see that her legs are together and beautifully wrapped by her dress again.
There are several shots scattered throughout the film where it's clearly noticeable that Robert Kazinsky is playing the role of Fili rather than Dean O'Gorman. This is especially noticeable in Trollshaw Forest, just as Thorin is beginning his song in Bag End.
The sharp end of Thorin's sword is lying parallel to his face, out of his reach on the boulder where the white warg flings him to. When the eagle flies Thorin off to the Carrock, it is carrying the sword gathered by his side with the handle pointing towards his face.
When the Company is being chased through the Goblin tunnels, several Goblins begin swinging at the dwarves from a bridge directly opposite them. When Thorin yells at Dori and Nori to cut the rope holding the bridges together, Dori's section snaps before he brings his sword down to cut it.
When the dwarves are seen around Bilbo's table before Thorin appears, Fili is sitting next to Kili on the right hand side. A shot later, Ori has taken Fili's place, as Fili is walking along the table handing out ales.
After Galadriel questions Gandalf "Why the Halfling?" in Rivendell, his hair changes of position some times: at the beginning he has a lock of fair falling by his left side, but later all his hair is slicked-back. Finally, a few hair cross his face, which are removed by Galadriel.
While the White Council is gathered in Rivendell, Saruman is being shot from his left side, where you can see that his hair is a bit 'wavy' but when the camera turns back to shoot right in front of him - you can see his hair is utterly smooth.
Early in the journey, as the Dwarves, Bilbo, and Gandalf are riding in the rain, none of the characters are really wet. In the downpour, water should be streaming down Bilbo's face and off of Gandalf's hat and beard.
After exiting the Goblin dungeon, when Thorin is thrown to the ground by Azog's warg, the Goblin Cleaver is clearly thrown to the ground, and is clearly out of reach and he faints. When the eagle grasps Thorin in his talons, the Goblin Cleaver is holstered with him.