There are a few different ways to become a hero in My Hero Academia, and quite a few ways to be permanently disqualified, but at least one major pro, the Rabbit Hero Miruko, had a surprising start breaking the law rather than enforcing it.
While she hasn't had much screen time in the My Hero Academia anime thus far, Miruko is an important player in My Hero Academia's horrific Paranormal Liberation War arc from the manga that's soon to be animated. Despite that, she's become quite popular by virtue of her costume design and spunky attitude, but that attitude is exactly what got her in trouble in the past, putting her dreams of being a hero in serious jeopardy. While Miruko's been fighting Shigaraki in My Hero Academia, her background, which is only elaborated on in the prequel/spinoff My Hero Academia Vigilantes, actually draws some interesting parallels to Bakugo.
In chapter 88 of Vigilantes, Miruko is introduced to what is known as the Underground Masquerade arc. At the time, she's a junior at an unnamed hero school in Hiroshima, on a field trip to Osaka, when she gets the sense of a fight going on nearby and splits off from her friends. Miruko quickly discovers the fight: an underground cage match, pitting masked fighters against each other with their quirks on full display. Since this is exactly what she was hoping to find, she throws together a costume and enters the ring as "Tiger Bunny," perplexing the referee and audience alike. Someone in the audience recognizes her from a rumor, however: a story of rowdy high school girl with rabbit ears, known only as "Usako," who became well known in the Hiroshima underground. And as it turns out, Miruko has not only already gotten in trouble for this--she was even arrested for illegal use of a quirk and suspended from school.
Miruko's presence is critical to saving the day, but she refuses any credit for it and rushes off before her absence is noticed, mostly because she fears the school will expel her for good. The incident is perhaps not the most surprising thing that could come from her past--Miruko has always loved the thrill of battle. More surprising is that a hero student could be arrested for illegal quirk use and still go on to become a pro, since quirk usage is so highly restricted that it's seen as a very serious crime. This goes double for Miruko, since she was explicitly using it to fight. As difficult as it is to get into a hero school, it seems like they're surprisingly forgiving of mistakes once there.
Fans may sometimes wonder how characters like Bakugo and Miruko manage to become heroes in spite of their violent outbursts, but this incident gives the impression that violent tendencies aren't disqualifying, and under some circumstances, can even be a useful trait for a hero. That's only become more true with the emergence of Nomu and other monster-like creatures in the My Hero Academia world, so a rough and tumble hero like Miruko might just be what's needed.