Continuity mistake: When the Cave Troll stabs Frodo, and then you see Sam close up and yelling "Frodo" He has 2 bloody cuts on his left cheek, and near his lip. But when the Troll is killed, and he slowly approaches Frodo, the cuts are moved further back, so they aren't as noticeable. But when Gandalf 'dies', and you can see Sam close up again, crying, the cuts have vanished all together. (The second timecode is when the cut vanishes all together).
New this month Continuity mistake: Upon the Fellowship entering the Chamber of Mazarbul and discovering the tomb of Gimli's cousin Balin, a ray of light is shown shining into the room from the side opposite to the entrance: the side with the long drop later featured prominently by Pippin. After exiting the chamber, another ray of light is shown shining into the tomb from the opposite side it was earlier depicted, appearing to enter the chamber from the entrance side through a small hole above and slightly to the right of the door. No such ray of light, or hole, is seen prior to the Fellowship entering the chamber: the direction of the light just completely flips upon them exiting the room.
Revealing mistake: In Balin's Tomb, after Merry and Pippin are separated from Frodo when the cave Troll smashes its club on the floor beneath them, the Troll first sniffs around the pillar to Frodo's right (viewer's left) as we see Pippin and Merry standing up, before the Troll comes around to Frodo's right and sniffs again. At the start of the next shot, we actually see Merry and Pippin in full view of their facial expressions, standing at the ready, with their daggers held high, continuing to just stare past the pillar behind Frodo. So, despite the fact that the huge Troll is appearing directly in front of them, to Frodo's right once again (before it roars), Merry and Pippin don't even glance in its direction.