Among the regular flavors of Zutara: Katara being kidnapped by 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.
Quite a few people like to write about how Zuko managed to find his mother.
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").