Among the regular flavors of Zutara: Katara being kidnapped or outright enslaved by (or for) Zukonote ironically, this was reversed by Enslaved, which is the most popular fic within this genre, Katara's season three hatred for Zuko revealing itself to be sexual attraction, Zuko siding with Team Avatar in the Crystal Catacombs, Mai dying in childbirth and Katara comforting Zuko, and Mai and/or Aang turning evil. Katara usually ends up as Zuko's queen in these stories (the flipside of this, Zuko renouncing his title and living at the South Pole with Katara, is rare).
A common plot in Tyzula fanfiction seems to be Ty Lee discovering that Azula is being neglected/abused in the mental institution and having her released. Generally, in these stories, Zuko will be completely oblivious because he's either never visited her or he just never noticed the bruises/how thin she was getting, Mai will be of the opinion that no punishment is severe enough for Azula anyway and repeatedly question why she wasn't sent to prison/stripped of her bending/executed, and no one except Ty Lee will believe that Azula can be redeemed (not without reason, but they'll be portrayed as in the wrong). This allows the author to write Ty Lee in a protector role in an us-against-the-world scenario.
The Dangerous Ladies, or just Azula herself, kidnapping or capturing Sokka, or they're stranded alone together in the wilderness, and they come to bond romantically.
If a fic set after the war isn't based around the gargantuan amount of reconstruction and peacekeeping required throughout the world, then it will casually mention it as if the reader should already know. Indeed, despite the series as a whole ultimately falling on the idealistic side of the sliding scale nearly every fan reaches the conclusion that no, it isn't going to be that easy. This, of course, became canon in the follow up comic series The Promise which shows problems include Fire Nation colonists refusing to move, lingering racism between nations and Zuko's own doubts about his leadership.
When an OC from the real world falls into the world of Avatar, one of two things are certain to happen: either she (it's almost always a she) will end up on Zuko's ship, join his mission, and fall in love with him or she'll end up joining the Gaang on their adventure.
There seem to be an alarming number of stories in which, after Zuko has his HeelFace Turn and leaves the Fire Nation, Mai is forced to marry Ozai instead.
It's all but universal practice for fics about the Fire Nation Royal Family to depict Iroh's wife (Lu Ten's mother) dying in childbirth and Zuko being born premature or an otherwise ill, fragile baby who almost died (extrapolating from the line "My father says I was lucky to be born").
Next Gen works were popular prior to TLOK revealing the canon offspring of the characters. Even then, some characters like Sokka, Suki, and Azula don't have any canonical children so OC kids are still commonplace for them.
Two characters, usually Katara and Zuko, are revealed to be reincarnations of Oma and Shu.
Zuko/Katara fan-works where the two either meet in disguise, or unknowingly fall in love, as the Blue Spirit and Painted Lady.
Characters being captured or otherwise sent to other nations as infants or young children, being raised amongst the natives, only to find out their heritage when they're older. Katara raised in the Fire Nation and Zuko raised amongst either of the Water Tribes are the most common usages.
Many fanfictions deal with Zuko telling the Gaang how he got his scar. Or rater them pestering him about it until he breaks down.
The Legend Of Korra
Seeing how the show ends with Korra and Asami entering a portal to the spirit world together, linking hands and looking romantically into one another's eyes, along with Bryke confirming that Korra and Asami fell in love over the course of the series, an incredibly common fanfic topic is exploring what happens immediately following the finale, namely Korra and Asami's vacation in the spirit world together.The Legend of Korra Turf Wars eventually showed what canonically happened.
There are many AU works where Asami is an Equalist. This comes from the Book 1 theory that Asami was an Equalist spy, which was debunked at the end of the season. However, Word of God did confirm that Asami was an Equalist during early development before becoming a Red Herring Mole. Most equalist Asami works are also Korra/Asami works. Usually, Asami will be sent by Amon to befriend Korra but will end up genuinely getting close to her instead.
Related to and often in tandem with the above are For Want of a Nail fics where the Red Lotus succeeded in kidnapping Korra when she was five, and raised her as one of their own. Additionally, any other possible members of the Red Lotus given there's implied to be cells operating all over the world.
Kuvira's past isn't given much depth in the show itself. All that's noted is that she was abandoned by her parents. She might have been taken in by Suyin as a foster/adopted daughter, but the wording is vague. As a result, many fan-works try to give Kuvira more backstory. Usually she was raised by Suyin but treated more like an apprentice than a daughter, which left her feeling rejected and didn't help her abandonment issues.
AU fics where Lin and Tenzin never broke up.
Stories set when Suyin, Lin, Kya, Tenzin, Bumi, and sometimes Izumi were growing up together are very common.