While it hasn't happened in the anime yet, Tomura accidentally kills his dog when his quirk first developed, as seen in a flashback. This happens in Chapter 236.
In the first chapter/first episode of the show, a character is suibaited, specifically told to jump from the roof of a building as part of a bullying scene. He rejects the idea.
Enji Todoroki towards Rei Todoroki. Season 2 in Shoto's flashbacks, he breaks her down mentally to the point where she pours boiling water on their kid because he looks too much like his dad during a melt down.
Shoto Todoroki is abused by his father, and his mother pours boiling water on his face during a mental breakdown. The clips appear in season 2 during the sports festival arc.
While the teachers talk to student's parents after the Kamino ward incident at the end of season 3, Bakugo is briefly slapped upside the head by his mom. It's framed as comedic.
Eri is abused by Overhaul in season 4, even though she isn't his real daughter. This is the worst case of it, as he preforms experiments on her and often breaks her apart and puts her back together. It involves physical, mental and emotional abuse. There's a lot of 'you're cursed', 'people will die because of you' and 'you belong to me' manipulation being thrown around.
In the manga, ch 234-236ish is Tomura's backstory. His father locks him outside whenever he talks about heroes and gets physically violent with him by striking a 5 year old in the face. When his quirk manifests he also tries steiking Tomura with a pipe looking thing from the yard.
Season 4, there's a character in the Yakuza who's quirk is based on getting sloshed. It makes other people feel tipsy and he's seen drinking on screen, but he only exists very briefly. It's not a long term thing.
one of the students in Class A has a quirk where he can talk to animals and bugs. although he usually finds the bugs too icky to talk to them, he asks for their help once during the training camp arc
During the training camp arc, there is a villain who uses a poisonous type of gas to attack the students. Many are seen coughing and struggling to breathe, some even pass out. Also, there is one character with a quirk that allows him to pass through solid object (such as walls or floors), to which during that time, he is unable to breathe.
Yes, a villain by the name of Dabi has a powerful fire quirk dubbed 'cremation', which he often uses to burn people alive, or attempt to. He has burns on his face and body, along with another character named Shoto (who has a prominent burn over his left eye). A 'hero' by the name of Endeavor also has a powerful fire quirk, and uses it against a mutilated monster called a nomu, a creature that was once human.
There are fight scenes in which characters are seen being buried by rubble, with implied deaths, but nothing major. Those that are shown survive, as far as the viewer can tell.
A man by the name of Grain is tortured to the point where one (or more (unclear)) of his fingers is chopped off 'to send a message'. This has not yet occurred in the anime.
SPOILER: Todoroki Shoto's mother becomes mentally unstable and pours boiling water on his eye. it is not directly shown in the anime, but I don't know about the manga
It is implied that the father of a young girl named Eri was reduced to nothing but a pile of clothes after his daughter accidentally uses her power on him. A young boy's parents, who were once heroes, were also killed in action, however, this is not shown on screen, merely told.
Fighting All Might during the exams, Bakugo vomits when decked in the stomach.
Uraraka also vomits before she gets a real handle on her quirk during the entrance exam.
Season 4 involves drug use by syringe. Gutsy Red Riot is the first appearance, when the white haired man holds up a gun is when it starts. There are themes of enhancement drugs for the first whole arc so they happen every now and again.
Recovery Girl also has a comically large one as a part of her outfit.
Todoroki's mother is kept in a mental institution. Shoto goes to visit her at some point, but the scene is just him talking to his mother and reconnecting.
Multiple hospital scenes. Students will be subject to numerous injuries in which they are required to get treatment for. Also, there is a moment in which numerous heroes fight in a hospital during a sting operation to find an evil doctor. This has not yet occurred in the anime.
No, but there is a character named Dabi who has burns that are self-inflicted, thanks to the nature of his quirk. However, it seems that these burns are unintentional, as are most (if not all) of the self-sustained injuries that characters have endured.
In the anime, there is a scene in which a character can control the walls around him, and uses his power to push the walls in towards the heroes, enclosing them in a very tight space.
Depending on the powers being used, and other anime effects there are flashing lights (electric powers, showing heroes with flashing lights around them to announce their presence/greatness, speed powers shown by rapid movement that causes the screen to flash, etc)
When Twice and Toga are talking to Overhaul in season 4 while infiltrating the base, Overhaul refers to Magne (canonically trans) as the wrong pronoun. Twice and Toga immediately correct him with "I think you mean SHE" and "Dont make that mistake again."
The League always refers to Magne as the correct pronouns and calls her 'Big sis Mag', they care about her very much. This is a one time occurance of misgendering, nobody makes that mistake again.
Season 4 episode 2, Magne dies. She's a confirmed transgender character. This isnt the only representation in the show of canon trans characters as we still have Tiger, but it's still a harsh and quick loss if you got attached to her.
There's a character naned Fatgum, but he's a pro hero who has a quirk involving uaing his fat as direct energy/being able to squish people into it to trap them. No jokes are really made about it, he's a respected pro hero.