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: As the group reaches the dilapidated house that Gandalf claims belonged to a farmer, a birdhouse on a pole can be seen on the threshold. In the next shot, it is gone.
Continuity mistake: When Bilbo brings the two bowls of stew for Fili and Kili there's a spoon in each of the bowls, but after Bilbo sets down the bowls one of the spoons has vanished when he picks up the bowls again in the closeup, but the spoon reappears when they run after the Mountain Trolls.
Continuity mistake: In the scene with the mountain trolls, the captured ponies are in a corral untied. Over the course of the entire troll scene, the horses go from being free in the stall to being tied up by the reins.
Continuity mistake: At Bag End when the Dwarves begin to burp, it shows one of the younger looking ones laughing next to another Dwarf and sitting down. The very next shot cuts to show that same Dwarf now standing and letting out a huge burp.
Continuity mistake: In the Fellowship of the Ring, Bilbo was old when he found the ring. In this film, he is young when he finds it.
Continuity mistake: The mark that Gandalf puts on the door (that looks like a capital F) changes. When Dwalin arrives, there is an additional part of a line on the second line of the capital F. But when Gandalf made it that additional part was not there.
Continuity mistake: In The Lord of the Rings, when the map of Lonely Mountain is shown, it says "Here of old was Thrain king under the mountain" (referring to an ancestor of Thror and not Thorin's father who has the same name and was never king under the mountain). But throughout The Hobbit, the map instead says "Here of old was Thror king under the mountain."
Suggested correction: Thrain was king after Thorin. The map would have been updated after Thrain's death, and he was subsequently buried there.
Thrain II was king after Thorin I and father of Thorin II (Oakenshield), but as said he was never king under the mountain but was king during their exile, Thror was the one who went back to Erebor to sit on the throne again until Smaug sacked it. The Thrain mentioned on the map is Thrain I (father of Thorin I), who is the founder of the kingdom under the mountain after Durin's folk were exiled from Moria by the Balrog (Durin's Bane, Durin being Thrain I's grandfather).
Also, even disregarding that, it is impossible for the map to have been updated to say a different dwarf's name, because the map is given to Bilbo at the end of the trilogy, and the ending shows a close-up of the map with "Thror" still written on it, whereas in the LOTR trilogy, where we see the exact same scene of Gandalf arriving at Old Bilbo's house (albeit from Gandalf's point of view rather than Old Bilbo's), Gandalf sees the map and it says "Thrain" on it.
Continuity mistake: After Bilbo has written the prologue to his book, Frodo is talking to Bilbo and then heading off to wait for Gandalf's arrival. This contradicts the extended edition of The Fellowship of the Ring, where Frodo is shown to have already left the house to wait for Gandalf before Bilbo begins writing his book (which he begins with a section called "Concerning Hobbits," which he states is "where to begin," indicating that Frodo left the house before Bilbo wrote any of the book).
Continuity mistake: At Bag End, Bilbo recovering from fainting, sits in an armchair. Gandalf is speaking with him. Behind Gandalf is an alcove with an unlit candle, which is lit in the next shot, although not illuminating any of the alcove, then it's unlit again.
