Continuity mistake: When the hobbits are hiding under the tree trunk from the Ringwraith in the beginning, you can see space to the left and right of the tree above them. Logically when the Ringwraith walks past the tree you would see it on the right side of the tree first, then on the left, but you don't - it looks like it walks out of the tree instead of behind it. [Confirmed on the commentaries - Elijah Wood asks his fellow actors if anyone spotted the mistake: 'It kind of magically comes out of the tree'. Sean Astin: 'You mean it doesn't pass from the other side?' Wood: 'No, it comes out from the centre.']
Revealing mistake: Once the secret door to Moria is open, when Gimli says, "Soon Master Elf, you will enjoy the fabled hospitality of the Dwarves," as Gandalf leads the way inside, note the glint of the trailing power cord as it moves along the ground behind Gandalf and goes directly under his robe. This is the staff's electrical supply cable.