A popular Alternate Universe is the idea that Midoriya would become a Villain rather than a Hero, inheriting All For One and often becoming a superior, if not someone competing with Shigaraki for dominance over the League of Villains. Often, he becomes Bakugo and Todoroki's Nemesis. That AU usually has Midoriya descending into insanity and becoming a very capable murderer of Heroes, though keeping his Hero Otaku qualities, usually he wears barman clothes akin of Kurogiri's and is a friend to Himiko and Dabi. There are enough fanfics with this plot to warrant it's own folder on the Fanfic Recs page of this trope.
One of the most popular fanfic ideas is to give Izuku the superpowers of a popular comic book hero like The Flash or Spider-Man separate from a Quirk, potentially using them in tandem with One For All.
Midoriya having a quirk is quite popular, and has spawned a numbers of variations to this plot:
A variant involves the Quirk being useful but not in the battlefield, like some kind of healing Quirk or sensory powers.
Other cases have Midoriya simply be a rare, late bloomer in terms of his quirk activating, and that it simply kicks due to a random event after his canon diagnosis as quirkless, or receiving One For All acts as a catalyst to awaken his initial quirk.
Midoriya receiving either his mothers quirk, (but significantly strengthened to the extent of full-on telekinesis), or his currently-unseen fathers fire breathing ability.
It has become somewhat popular to write fics where All Might is Midoriya's secret father or even that Midoriya is All Might's publicly known son. It's common to put this in Inko/All Might fan works, which is a ship with a sizable amount of fans. A minor variant of this is "All Might acting like Izuku's Dad," where the difference is that All Might merely acts like a father figure to Izuku, rather than outright being his father. This doesn't exclude stories where All Might becomes Izuku's stepfather, though.
While not as popular as showing him as a Villain, Midoriya as a Vigilante has some traction, specially after the Spin-OffVigilante: My Hero Academia Illegals was released to showcase the life of the Vigilantes. In some of these Vigilante fics, Midoriya may team up with Toga and Dabi who were given an Adaptational Heroism treatment. It's also semi-popular for him to form a Power Trio with Shinsou and Hatsume, particularly if he still winds up attending U.A..
Midoriya staying Quirkless and taking All Might's advice about becoming a police officer if he really does want to help people is a common plot that is quite popular in some circles.
A particular dark and angsty fanfic plot is one where Midoriya takes Bakugo's advice from the first chapter and commits suicide (almost always by jumping off the school's rooftop). How the story goes from there varies, but almost uniformly, they either follow Bakugo becoming guilt-ridden about causing this or Midoriya faking his death and becoming a member of the League of Villains. Sometimes, these two merge to form a story where a guilt-ridden hero in training Bakugo meets Midoriya, who he thought to be dead, as a villain, and tries to either turn him into a hero or is manipulated into becoming a villain.
"Dekubowl". There is a plenitude of fics where several characters compete for Midoriya's affection. Todoroki, Uraraka and Bakugo are almost always participants, but the entirety of Class 1-A are available contestants, as well as Mei, Shinsou, Toga, Shigaraki, Mirio and so on, truth be told, any given character introduced is a possible addition to Midoriya's fanon harem.
There are quite a few plots about Midoriya's childhood, in particular, how would his childhood go if someone else was his Childhood Friend instead of Bakugo. Usually, Shinsou, Iida and Todoroki are the ones used for these plots. Less common but also abundant, is Midoriya having a childhood friend with an OC.
Midoriya meeting All Might far earlier than in canon, receiving One For All, and allowing him more time to learn and master it far better than he has been doing, avoiding or lessening the crippling injuries he runs the risk of canonically.
Plots related to the characters' future as Pro Heroes are common subject in fanfiction, whether or not it includes shipping.
It has become popular to write about Eri's future after the Internship Arc, particularly, her relationship with Deku, Mirio and even Nighteye. Some believe she'll be adopted by either Inko, making her Midoriya's little sister, or Aizawa due to their similar Quirks.
Alternate Universe fics where Mirio and Deku are a couple sometimes end with them adopting Eri.
Due to Mineta's large hatedomdue to his perverted antics and Shinsou being a far more populartragic boy who wants to be in the Hero Course, several people who write fics prefer to simply write as if Shinsou is part of 1-A instead of Mineta and just forget he exists all together. Sometimes, when he is acknowledged, he is described as being widely disliked by his classmates who are glad he left and receive Shinsou with open arms. When Mineta is even acknowledged, the typical way of replacing him with Shinsou is to have Mineta finish last instead of 2nd to last in Aizawa's quirk apprehension test on opening day and thus get kicked out of the hero course, since the only reason Midoriya didn't suffer this fate in canon is that he impressed Aizawa with his combination of resolve and cleverness. Alternatively, Aizawa learns of Mineta's sexual harassment and creeping on the girls of 1-A and swiftly moves to expel him.
A trend in fanfiction is the idea of Hisashi Midoriya, Izuku's father, being a Villain, expanding why he hasn't shown up or take an active role in being a father figure to Izuku for years. This follows from the theory of Dabi possibly being Izuku's father but instead of using the theory completely, it changes the theory that Dabi is a different person from Hisashi. It has become especially popular to have Hisashi and All For One be the same person.
Something that is even used by Horikoshi himself in the side materials is to make Villains with weird Quirks that can kick-start a plot by affecting one of the main characters, the author used it in the "Smash!" comics to make a story about Bakugo acting as classic BishōnenLarge Ham due to a Villain's Quirk. In fanfiction, this is often used in PWP with a Villain having a lust-inducing Quirk. This was even the plot of an OVA, as there was a person with a zombie-making Quirk.
A popular villain quirk to make fics about are ones where they turn Midoriya into a girl. More rarely, this sometimes happen to Bakugo.
Group chat fics, which are Script Fics where the framing device is that they're a transcription of internet chatroom logs or text group logs of the main characters. Like real life group chats, they often tend to be heavy on meme humor and light on plot.
CrossoverFanfics where Midoriya meets and fanboys over other known heroes also exist. Things like meeting Superman, The Incredibles, Spider-Man, etc are common plotlines. An entire block of most crossovers with the latter often entail Spidey being dropped into the world of My Hero Academia to become a student in U.A. Other depict these more experienced heroes as members of American agencies who often are depicted as being some of the most experienced pros in the world. In the event most characters like the aforementioned Spidey do not get an Age Lift to match the ages of the U.A. students, their younger Legacy Character counterparts (Miles Morales, Robbie Reyes, Kamala Khan) take the spotlight instead.
Some people like to create stories about Eri and Kouta becoming U.A. students when they grow up to be teenagers.
Bakugo having a Hair-Trigger Temper is canon, of course, but many stories have him erupting and doing something terribly awful because "stupid Deku" committed the grievous crime of existing one time too many. What happens afterwards tends to be dramatic more often than not.
For whatever reason, many authors like to pretend all the adults are in the same age range and went to school together. While this is canon for Midnight, Eraserhead, and Mic, putting All Might and Endeavor with them is quite a stretch.
Because of the sheer ridiculousness of the medical test (so apparently an extra joint determines the chance you'll grow a tail), some stories have this practice either be a sham, or an archaic way that isn't supposed to be used anymore.
In Alternate Universes where Deku doesn't inherit One For All, the most common person to inherit it in his place is Mirio, as All Might was going to meet him to see if he was worthy before his chance encounter with Deku.
Melissa Shields inheriting One For All is also gaining popularity.