With One-Punch Man because of how both series are Japanese takes on the Western superhero genre. This has created a significant overlap in the fandoms rather than a rivalry. The fact that One-Punch Man is a seinen with a much older target demographic and more adult humor than My Hero Academia helps. Heck, it's to the point where some jokingly suggest that One-Punch Man is a Distant Finale for My Hero Academia, with Saitama being the final/current user of One For All. Both of the series have also very much well liked takes on the Hero Killer concept, with Stain or Shigaraki being paired with Garou a lot.
The fandom is on friendly terms with ONE's other series Mob Psycho 100 due to both shows featuring socially-awkward but powerful kid protagonist. There are many fanarts and fanfic where both protagonists befriend each other or Mob himself being depicted attending U.A. to train his telekinetic powers.
There's one with Naruto. Horikoshi has cited Naruto as one of the inspirations for the series, and it started around the same time Naruto ended, so the timing was good for fans to find a new series to fill the void. This is exemplified by this fan-made comic.◊ Now with an updated version in color.◊
Both My Hero Academia and Sailor Moon have a great friendship. Additionally, there are some fans who wished to have Midoriya and Usagi show up in some crossover fighting games like BlazBlue: Cross Tag Battle.
A small one with the Disney cult movie Sky High, since the manga is basically a Japanese version of it.
It has one with Tiger & Bunny, due to both being anime series about Western-style superheroes.
It has a pretty big one with Little Witch Academia, largely due to similar ideas behind the plot, youngster with a dream and a natural lack of skill for what they want to do (Midoriya and Akko) gets to follow in their hero's footsteps (All Might giving Midoriya One For All, while Akko receiving guidance and the Shiny Rod from Ursula) at an academiaof adventure where said hero teaches. A lot of crossover fan art of Akko in U.A. and Midoriya in Luna Nova was prompted soon after Little Witch Academia's airing ended.
Fans of Bungo Stray Dogs are friendly with My Hero Academia fans and/or have watched My Hero Academia as well due to the similar setting of powers though the former are more in line with the supernatural. It is another common fandom for crossovers in fan works.
There seems to be one with Dragon Ball, as indicated by the fact that the majority of "You Say Run Goes With Everything" videos are Dragon Ball-related.
There's one with JoJo's Bizarre Adventure, due to how many references and similarities there are. Like with Dragon Ball, a lot of the "You Say Run Goes With Everything" videos are synced to moments from JoJo.
With Haikyuu!!. There are actually several voice actors who voice significant (whether they're main or popular) characters on both shows.
Among Western YA reader's there's some overlap between this series and the Renegades Trilogy by Marissa Meyer due to having similar settings and world building. Even the same criticism's of a world where superheroes run rampant. Nova's motivations and criticisms of her world are remarkably similar to Shigaraki's.
With Invincible, due to the similarities between Midoriya and Mark and both being based on Western Superheroes.
◊ Now with an updated version in color.
