Continuity mistake: When Azula is asking Zuko whether or not he thinks the avatar may have survived, Zuko goes into a flashback. When he snaps back to real time, Azula has her hands behind her back. When the shot cuts, her hands are instantly by her side.
The Headband - S3-E2
Continuity mistake: When Zuko and Mai are having a picnic, Azula asks Zuko if she can have a word with him. In the next shot, Mai is wearing a black glove on her left hand, which she wasn't wearing in the previous shot. Two shots later, the glove is gone. (00:15:03)
Continuity mistake: Toph mentions the worst thing about being disguised is they don't get the hero worship anymore and she misses the love, to which Sokka grumbles, "Boo-hoo, poor heroes." Then, as Sokka says, "It's just, all you guys can do this awesome bending stuff," the triangular emblem on Aang's red "headband" has vanished, and in the next shot, it reappears. (00:03:35)
Continuity mistake: After Sokka admits to Piandao that he thought he messed up everything they worked on, when Piandao says, "You messed things up in a very special way," the yellow trim at the front of Piandao's collar is incomplete. The yellow trim is rendered properly when Piandao says, "Your sword must be an extension of yourself." (00:13:55)
Continuity mistake: When Piandao tells Sokka that creativity, versatility, and intelligence define a great swordsman, he holds out Sokka's sheathed sword with its hilt at Piandao's right hand. However, in the next shot facing Sokka, the sword in its scabbard now lies on Piandao's hands the opposite way, with its hilt at Piandao's left hand. The position of Piandao's hands and thumbs changes as well, between shots. (00:17:25)
Continuity mistake: In the shot after Mai says "Saving the jerk who dumped me" you can see the rods which the prison guards used to slow down the mechanism of the gondola are nowhere near the wheel. (00:19:55)
Continuity mistake: When Azula says: "What is she doing?" Ty Lee is standing behind her. When the shot cuts and Ty Lee shrugs, they are standing side by side,. (00:20:30)
Continuity mistake: When Zuko goes inside Sokka's tent to talk to him, Sokka is lying on the floor, but in the next shot, he is a few inches forward from where he was last. (00:07:24)
Continuity mistake: When Katara is sitting on the dock at the end, she is wearing her blue gloves on her hands, but then when she stands up and walks over to Aang, she is not. (00:22:26 - 00:22:52)
Continuity mistake: When Katara goes to hug Zuko at the end, she is not wearing gloves on either of her hands, but then a glove appears on her left hand mid-shot, and her right hand also has one once she puts her arms around Zuko. (00:23:04)
Sozin's Comet (1): The Phoenix King - S3-E18
Continuity mistake: When Aang is waking up on the giant lion turtle, he is not wearing shoes in the first shot, but in the next shot, he is. (00:22:35)
Sozin's Comet (4): Avatar Aang - S3-E21
Continuity mistake: Throughout the entire show, Aang's eyes are grey, but in the last three episodes (Sozin's Comet Part 2, 3 and 4) they are suddenly brown.
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.