Among the regular flavors of Zutara: Katara being kidnapped or outright enslaved by (or for) Zukonote This was reversed by Enslaved, 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).
Similarly, 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 Heel–Face 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").
AUs where Katara and Zuko are forced to marry to end the war.
Fanworks where Aang has a Long-Lost Relative somewhere aren't that uncommon. Usually it's an OC but sometimes he's instead related to Ty Lee through the fanon Air Nomad in her.
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.
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.
Related to and sometimes 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. This is frequently combined with the above Equalist Asami AU.
Zuko and Katara hooking up as seniors.
Kuvira's past isn't given much depth. 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, and Bumi were growing up together are very common.