Continuity mistake: When Anakin is sitting in his Naboo starfighter about to take off, there is one shot of R2 with his projection nozzle on the left side of the red light instead of the right.
Continuity mistake: When Qui-Gon delivers the hyperdrive, the way he holds the reins of the eopie he's riding changes between the first two shots. (01:08:25)
Continuity mistake: When Anakin and Padmé arrive at the Podracer hangar, Anakin is taller in the first shots than in the shot where he gets off the animal they're riding on. (00:51:45)
Continuity mistake: When Amidala's starship runs the blockade, after the spacecraft flies towards the screen, there's a shot where R2-D2 is missing from the top of the ship. (00:25:35)
Continuity mistake: As someone else has noted, Obi-Wan's braid changes sides a few times in the movie. In the first scene with Jar Jar, his braid stays on the correct side of his head, but after he says, "If they find us, they will crush us" it goes from hanging in front of his shoulder to hanging behind it. (00:12:05)
Continuity mistake: When Padmé is to come with Qui-Gon after the Queen's ship has landed on Tatooine, he says, "This is not a good idea", and something blurry flies by in the upper right hand corner. It is not there before or after. (00:29:25)
Continuity mistake: Jar Jar knocks down a bunch of droids in Watto's shop, but when they leave, you can see the feet of the now stacked up droids. (00:33:15)
Audio problem: When Queen Amidala shows the holographic map of the palace and discusses the plan for sneaking into it, you can tell that her explanation of the plan is a recording. There is one shot where you can see her face and her lips are not moving.
Other mistake: During the fight scene in the Theed hangar, several times the droidekas (destroyer droids) are shown shooting, but no lasers come out of their guns. (01:46:40)
Continuity mistake: When the final duel between Obi-Wan and Darth Maul begins, they are fighting in a particular area next to the pit. But when Obi-Wan cuts Maul's lightsaber in half, the background changes and they are suddenly fighting on a different side of the pit. (01:56:40)
Continuity mistake: During the funeral pyre scene, the folds in Obi-Wan's hood change between consecutive shots.
Continuity mistake: When Palpatine and Queen Amidala address the Senate, the holocam flying around near them is inconsistently placed between shots.
Continuity mistake: In the first scene in Watto's shop, a cloth lying on a device on his desk keeps changing position between shots. (00:30:25)
Continuity mistake: When R2-D2 and three other astromech droids fixing the Nubeing ship, Jar-Jar is talking to them and there is a black one. This black one turns into a blue one when they are fixing the ship. It's the second one to get blasted.
Continuity mistake: When Obi-Wan falls off the walkway during the duel with Darth Maul, his lightsaber lands on a lit strip of the floor. A few shots later he jumps up and retrieves his saber, which is now resting on a dark area on the floor. Even if the saber continued to roll on the floor between the shots, it couldn't have ended up in this position. (01:50:50)