Continuity mistake: When we first see Alfrid with the Master of Laketown, he is talking about the rebellious people. On his left shoulder is a big splat of yellow goo, probably from a thrown egg. In the previous scene, where he confronts Bard and tries to have the barrels of fish dumped in the water, he hasn't got the yellow goo on his shoulder yet. In the extras of the Extended edition it is revealed that there was a scene filmed where Alfrid was pelted with eggs by the townsfolk when he walked to the Master's house, but the scene was not included in either the theatrical or the extended version. The egg also explains his line: 'the mood of the people is turning ugly'.
Continuity mistake: When Bilbo and the dwarves leave Beorn's house, it is just past daybreak, with the sun presumably rising in the east, their direction of travel. Later in the day, when the band realize they're lost in Mirkwood Forest, Bilbo says "We have to find the sun," and he looks upward, spotting the sun at about high noon. But when he climbs the tree and pokes his head through the canopy, he looks east and it is daybreak again with the sun just rising. Where did the day go?