The author has a great deal of these in delving into the characters' psyches. For example, Zuko has had a hard life, while Aang is unused to having trouble with things, leading Aang to slack off when he runs into trouble with bending training, and for Zuko to show him little sympathy for doing so.
Arc Stall: Most of Not Stalking Zuko, the longest installment in the series, takes place on Ember Island, and it takes almost half that fic to get up to "The Ember Island Players."
Myth Stall: Zuko and Katara are hesitant to go forward with their romance for several reasons, including that the war is still going on. When Katara finally makes an Anguished Declaration of Love, Zuko soon forgets it as a result of the trauma of getting hit by Azula's lightning in their Agni Kai, and Katara doesn't try again until near the end of the series.
Aang. Some agree with the author that he was wrong to run away in the Grand Finale, and should have had to own up to his mistakes, give up on pursuing Katara and learn other lessons he never did in canon. Others believe the author isn't being fair to Aang (for example, since he essentially got kidnapped by the lion turtle, the author shouldn't accuse him of running away), and that essentially saying he has to kill Ozai reads like a Family-Unfriendly Aesop.
Mai. Some argue that the fic treats her relatively well while still showing why her relationship with Zuko wouldn't work out. Others believe that she's suffered Flanderization into an even more snarky jerk who essentially uses emotional blackmail to restart her relationship with Zuko, and that by showing the Maiko relationship to be utterly dysfunctional, the author fails to consider that she and Zuko were largely happy together.
Jet. The main point of contention is not the fact that in the fic, he didn't die under Lake Laogai, but that he's representative of the more "cracky" parts of the fic, and whether or not readers like his portrayal here depends on whether they think those parts are good.
Hakoda. The fact that he's not fond of relationships between the Water Tribe and Fire Nations leads him to initially oppose Zuko getting together with Katara and Ming with Bato, to the point of suggesting to the Fire Nation halves of those couples that they break up. Some fans understand his perspective, even if they don't agree with him, while others find the way he goes about dealing with the situation to be entirely inappropriate (granted, he does get called out on this in the story). His initial unwillingness to support Katara's desire for more women's rights in the Water Tribe or even understand where she's coming from gets a similar reaction- some conclude that he naturally would have a hard time understanding how much Katara's changed while he was away, while others see him as almost as sexist as the rest of the men in his tribe.
Not only are viewers divided on whether the author treats Aang and Mai well, but the same goes for the author's refusal to accept Kataang and Maiko as viable pairings. Some believe that the author correctly points out problems with the pairings that canon failed to address, while others believe that the author doesn't give them enough credit or properly acknowledge why they do, in fact, work.
The length of the author's notes draws a certain amount of controversy. Some people enjoy hearing the author's thoughts, while others wish that she'd shorten them or post them elsewhere.
The pacing is also divisive. Some readers are highly critical of how long it takes for Zuko and Katara to finally acknowledge their feelings for each other, and believe that all the side plots only drag out the romance arc while adding little to the fic. Others, however, are glad that the romance isn't rushed when it wouldn't be in character for either of them to quickly get together, and like the worldbuilding and Character Development for the rest of the cast.
Contested Sequel: Not Stalking Firelord Zuko. While its popularity is comparable to the other two installments, and people like it for the same reasons, it also has drawn criticism for the pacing and portrayal of some characters, like Mai and Jet.
Creator's Pet: The author says that Aang was this for Bryke, and as an attempt to compensate for this by having him undergo hardship, being proved wrong, and not getting together with Katara, which makes him come off as the inverse of this trope.
Die for Our Ship: A less extreme case, but Aang's portrayal in this fic falls into the first(Aang is shown slacking off or wanting to do more frivolous training exercises, and the author openly states in Chapter 6 of Stalking Zuko that she believes that he got Katara as "a prize for being a good boy"), third(suggesting that Aang and Katara have a "maternal" relationship), fifth(suggesting that Aang is a hypocrite and he's trying to deny Katara the chance to obtain personal closure) and sixth (Aang's adherence to Air Nomad culture is treated as a flaw, and his need to re-establish the Air Nomads is said to preclude a relationship with Katara) bullet points on Avatar's Die For Our Ship page.
Esoteric Happy Ending: The author and Zuko's opinion on the story of Oma and Shu. Oma ended a war and founded a city, but Shu, the man she loved died before she could do that. Since Lu Ten's girlfriend met a tragic end because of her grief over Lu Ten's death, Zuko understands how painful losing a lover can be.
Fanon Discontinuity: Avatar: The Last Airbender – The Promise has been disavowed by the author as she believes that it makes no sense and puts everyone out of character. She admits the series is a good idea but the execution is awful. To varying extents, the rest of the post-Avatar works (the comics and The Legend of Korra) have also been rejected in favor of the author's fanfic, for varying reasons (for example, she doesn't want Aang to get together with the girl he had a crush on when he was 12, nor does she want him to have three kids, only one of whom is an airbender).
Fridge Brilliance: The Earth Kingdom has different attitudes and better understanding of psychology. If the Dai Li are able to have the most sophisticated mind techinques, it implies that they have a good understanding of psychology and how the mind works.
Which is why in Not Stalking Firelord Zuko, Azula doesn't have a true therapist as Dr. Yang has no experience with psychology while in Jet's Troubling Obsession Smellerbee, Longshot and Zuko were able to send Jet to a psychologist.
One that's pointed out by the author, but in Not Stalking Zuko, the scene breaks have a ? between two hyphens, rather than a !. This shows that Katara is questioning Zuko, rather than angry at him.
Fridge Horror: Untrained firebenders in Sozin's comet had their powers increased to dangerous levels...
What kind of power did Ozai to disappear a public figure like Princess Ursa?
The Joo Dees
Especially since it's revealed that Ursa is a Joo Dee.
Fridge Logic: "The Boy In The Iceberg" does a somewhat accurate rendition of Combustion Man's fate (blown up after getting a boomerang to the head), save for the general dramatizing of the scene, such as having Combustion Man give a Final Speech when he never talks. One has to wonder how they knew that if only the Gaang saw the final battle with him.
Hilarious in Hindsight: It accurately predicted the tension between the Northern and Southern Water Tribes a year before season two of The Legend of Korra, down to how they viewed each other. This is further reinforced upon the release of "North and South," when Gilak is especially bitter about how the Northern Water Tribe didn't help while the Fire Nation invaded the south, and is imposing its culture on the South now that there's peace- in the fic, the South also has that complaint, while there's tension as a result of the North trying to persuade them to adopt their culture.
Informed Wrongness: Aang gets treated fairly harshly by the narrative and the other characters for letting Ozai live, forgetting that 1)he succeeded in stopping Ozai from destroying the Earth Kingdom, and 2)that it's possible to execute Ozai or imprison him while cutting him off from any possible supporters.
Ron the Death Eater: Aang gets portrayed as being highly selfish and immature. He pursues his one-sided crush on Katara and is portrayed as not knowing how to do so without forcing his feelings on her. He's also portrayed as putting Air Nomad beliefs above all else, and is portrayed completely unsympathetically for doing so.
Tear Jerker: Katara having no idea what to do after Zuko is electrocuted and her nearly having a Heroic B.S.O.D. over it. Just her helplessness about it.
The entire chapter Deeper Scars.
Unintentionally Sympathetic: Aang. Katara and the author often take the least favorable interpretation of his actions and motives, such as suggesting that he deliberately ran away and spared Ozai only to keep with Air Nomad teachings. While Katara claims to view Aang as a friend, she often seems to view him with a great deal of pity and reluctantly interacts with him, even when it seems as though he doesn't fully deserve that treatment. As a result, while some viewers believes this is necessary for Aang's Character Development, others believe that the story is too harsh on him.
Unintentionally Unsympathetic: Toph's parents are said to love her, but are very overprotective of her. While they do come around, Poppy's first letter to Toph essentially belittles her, accuses her of being selfish and plays the victim card to get her to come home, which borders on emotional abuse.