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)
Bato of the Water Tribe - S1-E15
Other mistake: When Katara is behind Sokka and he is speaking she leans back and as she does her eyes stay in the same place while her head moves, then it fixes itself. (00:16:20)
Continuity mistake: When Katara is trying to heal the wounded earthbender, she gives Sokka the idea to take down The Drill from the inside. Sokka then stops her from saying anything while she is sitting down, and she is wearing her necklace. When she stands up in that same shot, the necklace is gone, like they forgot to animate it. (00:07:35)
Other mistake: When Aang emerges from the water on the giant sprout in The Avatar State, the arrow on his head is not glowing.
The Tales of Ba Sing Se - S2-E15
Other mistake: The Tale of Sokka: When Sokka delivers his final haiku to the girls, he inadvertently adds an extra syllable to the final line. In the close-up of him counting his fingers, the iris of his left eye is rendered over his hand. (00:14:42)
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)
The Crossroads of Destiny - S2-E20
Continuity mistake: When Aang enters The Avatar State, the arrow on his head is glowing, as it should be. But in one shot showing his backside as he rises into the air, the arrow is not glowing. (00:19:44)
The Western Air Temple - S3-E12
Other mistake: When Sokka and Aang put Toph's feet into the fountain, Aang's hand (the one holding Toph's left leg) is incorrectly drawn with a blue glove on it. The glove is gone in the next shot. (00:15:29)
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)
Other mistake: When the group is first shown inside the store, Sokka starts running forward. Toph stands behind Sokka, but as Sokka is running away from them, she is layered over him. (00:04:43)
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: 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)
Other mistake: When Aang is reading the letter from The Guru, as Sokka says, "What's a guru? Some kind of poisonous blowfish?" his right eye loses its white colour three times during the shot. (00:18:19)
Continuity mistake: After Meng gives Aang the tray of food and Sokka takes it from him, Aunt Wu shows up, and it shows a shot facing her front. In this shot, Katara's back braid is missing; then it reappears a moment later. (00:07:05)
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)
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)
Other mistake: When the group goes to hug Sokka before he and Aang go to Chameleon Bay, you can briefly see that Toph is incorrectly drawn with arrow tattoos on her hands, like Aang has. This happens very fast, but you can still catch it when watching the episode normally without pausing or slowing it down. (00:20:35)
The Firebending Masters - S3-E13
Other mistake: When Ran and Shaw (the dragons) breathe the colorful fire around Aang and Zuko, and the camera moves down to them, Zuko's scar is on the opposite side of his face, and his sword and Aang's clothing are worn over the opposite shoulders in a mirrored shot. (00:20:01)
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)
Answer: The pirates are from the episodes "The Waterbending Scroll" and "The Waterbending Master" (They're the ones Katara stole the scroll from). The traveling musicians are from "The Cave of Two Lovers. The prisoners of war likely refer to the Kyoshi Warriors, the men of the Southern Water Tribe, and possibly Bumi and the other citizens of Omashu.