Continuity mistake: In the scene where Bilbo spares Gollum's life, Gollum turns around facing the invisible Bilbo and extends his right hand out to a rock. The far shots from in front of Gollum show his arm almost fully stretched out and his hand away. But the shots from behind Gollum show his right arm bent on the rock with his hand much closer and level with his head.
Continuity mistake: During the song "Blunt the Knives", Balin is seen in a shot from behind sitting at the end of the dining table. He is shortly afterwards seen dancing with a broom two doorways behind. The next shot then shows him sitting down again, bouncing plates off one he is holding.
Continuity mistake: Shortly after Bilbo finds the Elvish sword that turns blue in the presence of Orcs and Goblins there's a fight with Orcs at the entrance to the hidden valley. However, the sword doesn't glow blue. In subsequent fights with Goblins and Orcs the sword does glow blue, even when sheathed.
Other mistake: Just after he arrives at the gathering Thorin states that he wouldn't have found Bag End had it not been for the mark Gandalf made on the door. Bilbo then claims that 'There is no mark on that door, it was painted a week ago'. However in the earlier scene where Gandalf makes the mark the paint work is cracked and weathered, more than it would have done after only a week.
Continuity mistake: When the company are surrounded by Wargs outside the Hidden Pass, Thorin can be seen holding his sword Orcrist in one hand and an axe in the other; the shot changes and he is then holding his sword with two hands, the shot changes again and he reverts back to the original configuration.