With such an expansive cast and interesting lore, it's no wonder My Hero Academia has as many fanfiction stories as it does. On Archive of Our Own alone, MHA has over 21,000 works for fans to peruse. The majority of these fall under the Alternate Universe (AU) genre, where characters from the canon storyline are brought into other worlds by writers for fun and exploration. In MHA's case, it could be a place where Quirks don't exist and the characters are regular high school students, or where Class 1-A's students are villains instead of heroes in training.
Whatever the case, this genre of fanfiction is usually created to explore the world and characters in ways the canon doesn't. Because AUs can be about anything, their content is varied. However, it can be broken down into a few general categories.
The most popular Alternate Universe fanfiction in My Hero Academia revolve around Deku. They can take place in worlds where Deku becomes a villain or a vigilante, or where he has a Quirk or does not inherit One for All. There's at least one prominent fanfiction for each of these ideas, but one takes the world of MHA and looks at it from a unique perspective. In The Roundabout Way of Fate by itsmyartfam, Deku is born with a powerful healing Quirk that nobody was expecting. As he ages and eventually gets to UA, Deku carves out a place for himself in the world of healers and discovers it may be harder than being a hero.
In this fic, the author considers what healers may have to handle in a hero society -- from the dangers they face to the politics involved -- and builds a story around it. It still follows the events of MHA but tweaks them with the changes to Deku's character in mind, so every story beat is different from canon. It adheres to the official material as much as possible, but not all AUs do this.
One of fanfiction's main draws to readers is wish fulfillment. Fans often develop ideas about a character from canon context that they want to explore on a deeper level. Especially when this aspect of a character isn't utilized in canon, AUs are created so fans can get what they want. A popular change to canon involves Aizawa Shota and the canon fact that he is essentially fostering Eri while she learns to control her Quirk. To fans, this shows a rarely seen parental side of the hero. As a result, fanfiction where he has adopted any number of characters, especially Shinso Hitoshi, are a common sight.
Of Owning a Cafe and Other Heroic Deeds... by Beware_The_Tristero takes this idea to the extreme in a cat cafe setting. In a world where Aizawa lost his parents as a child and then aged out of the foster system, he never went to UA or pursued a career in heroics. Instead, he seeks to improve the area he calls home with a community hub of his own design. Thus, the Noraneko Cafe is born. Along the way he takes in a couple of troubled kids, becomes the focus of a group of Pro Heroes, and gradually changes his neighborhood. While Quirks still exist, everything else from character occupations to ages has been altered to fit the author's vision.
Besides wish fulfillment, many people write fanfiction as a safe way to escape reality or cope with life. Bakugo Katsuki in particular is an interesting case study in this realm. In canon he's abrasive, loud and as explosive as his Quirk. Because of this, some MHA fans hate him and find him unbearable. There are others, however, who adore him -- and they seem to be the majority based on official popularity polls. One reason for this is because those who are victims of abuse see pieces of themselves in him. His unwavering independence, enforced distance from others and near-constant anger can be interpreted as responses to trauma.
As a result, writers create fanfiction exploring this with varying degrees of maturity. On one end of the spectrum are the AUs where an adult, usually a Pro Hero, in Bakugo's life notices a problem and saves him from what is happening -- like Best Jeanist does in Lessons Learned by Sif (Rosae). After seeing his mother's treatment of Bakugo first hand, Jeanist takes emergency custody and works to remove him from his home.
On the other side are stories where Bakugo saves himself by running away. One example of this is Healing Hands by KayMacBurn. In this AU, a middle school Bakugo is frequently kicked out of his home for long periods of time and starts hanging out in poorer areas of Mustafu. There he meets a doctor who treats anyone who comes into his clinic, and the man pseudo-adopts Bakugo by making him his assistant. Under his care, Bakugo gets familiar with the city's underbelly and uses his experiences to fuel his desire to become a hero.
Whether it's to see the canon world in a new way or for deeper reasons, Alternate Universe fanfiction is just as varied as the people who write it -- and My Hero Academia is no exception. For this reason, it's very likely there's something out there to please every reader.