Katsuki Bakugo is probably the most divisive character in the entire franchise. Despite continuously winning popularity polls, Bakugo's haters are in about equal number:
For those who like him: Many fans love him for being a very well-written character, often calling him one of the best written characters in Shounen as a whole and maybe the best written rival (possibly only losing to Vegeta); for his badassery in the fights he partakes in, such as his fights with Midoriya, Todoroki and Uraraka; and for his Jerk with a Heart of Gold personality where, despite his rudeness (which is often played comically and many call him one of the funniest characters in the manga), he is shown to care for his classmates, especially Kirishima. Helping that, he is very handsome and is a Launcher of a Thousand Ships only behind Midoriya himself in the number of ships, with a particularly large female fanbase.
As for those who dislike him, quite a few feel like Bakugo is still an unlikable jerk in spite of the reasons given to like him, and the fact he hasn't apologized for or even acknowledged his time bullying Midoriya for so many years even after undergoing Character Development is quite the sore spot for some fans. The reasoning given for his horrible behaviour is also very weak, and he doesn't have a Freudian Excuse that can explain why he wants to maim a guy who just wants to be his friend and was never anything but nice to him, even despite his violent hostility. Even further, given his habit of striking Midoriya in a rage (and Midoriya never really seeming to care) and flat out launching deadly attacks at him with a flimsy excuse in the Battle Trial Arc, along with his other massive freakouts that lasted at least until the Provisional Hero License Exam Arc. Many fans see him as little more than half a villain already, who is only one good push (or one failure to dodge) away from becoming a full-fledged murderer.
There are also a number of people who dislike Bakugo for meta reasons, rather than any reason within the text itself. Among them are people who dislike him based on the fact that his fans are the most rabid and aggressive in the fandom, even going as far as bullying people who don't think he is the best character in the manga. Another set of people believe that Bakugo receives favoritism from Horikoshi (who admitted to liking Bakugo more than Midoriya) and hate him due to Midoriya's seemingly never-ending worship of him, complete emotional indifference to Bakugo flat out beating on him and attempting to harm him, and constant yammering about what an "amazing" person Bakugo is simply because he has a powerful Quirk and a determination to be the best. To some, it's almost as if Horikoshi is using Midoriya as a mouthpiece to shill his seemingly favored character, which doesn't help a general belief that Bakugo gets a certain amount of undeserved Character Shilling and is prone to not facing many (or any) consequences for his actions.
Despite his cool design and interesting skillset, Shota Aizawa/Eraser Head has led to a few divisions among fans:
Those who like him see him as a realist who is only harsh in the way that he is because he wants to prepare his students for a career path fraught with constant danger. His Kick the Dog moments are tempered into Jerkass Has a Point, rightfully pointing out that Midoriya's Quirk is Awesome, but Impractical and could cause problems when a part of a team, and even then his tendency to focus on Midoriya is because no one thought to inform him about the true nature of his Quirk. Some of his defenders also acknowledge his flaws, as mentioned below, but contend that he isn't meant to be seen as a perfect or even very good teacher.
Those who dislike him, especially in the West, think that his methods and his belief in The Spartan Way borders on unreasonable. He acts as though he can guess if a student is a prodigy or a lost cause on the first day based on whether they can throw a ball and touch their toes in a flashy way, expecting his students to have 100% control of their powers in a society that forbids them to use their Quirks in public areas. He has also neglected to teach them important facts for high-stress situations under the assumption that they'll figure it out at the last second and neglected to give his students proper psychiatric help after they are forced into many life-or-death situations on his watch. Additionally, it has been pointed out that in a society like Japan, his tendency to expel students at the drop of a hat are not only cruel, but can be incredibly devastating to the expelled students in the long run. Even after it's revealed that Aizawa re-enrolled the expelled students, one of them points out that the expulsions will still be on their permanent records. Some people also dislike him for threatening to expel almost the entire class over five of them taking part in an unauthorized mission to rescue Bakugo, since 1) the mission not only succeeded, but also did not involve any direct combat with the villains, and 2) rather than willingly turning a blind eye to their classmates' actions, the other students didn't necessarily know that their classmates actually went through with the rescue until later (as Tsuyu points out).
To some people, Tomura Shigaraki is a weak, ineffective and incompetent villain whose Psychopathic Manchild shtick annoys rather than terrifies, especially in early arcs where his attempts to attack the main characters were mostly pushed forward by All For One than by his own hand. Others, however, enjoy the fact that, unlike many other villains, Shigaraki starts as weak and inexperienced as Midoriya himself, but develops both a good relationship with his companions, grows as a leader and becomes a more powerful fighter, not to mention his sad backstory revealed later. They believe all of these make him an Evil Counterpart of Midoriya, and thus the best choice for an antagonist to oppose him.
Originally an Ensemble Dark Horse for the beloved arc he debuted in and for presenting some valid criticisms of hero society, Stain began to fall out of favor with a significant portion of the fanbase after a deeper introspection of both his character and ideals, particularly after his origins were explored in the Vigilantes spin-off. Supporters view him as an Evil Is Cool character that raised some valid issues and was right in some of his beliefs, as the narrative itself supports. Detractors, however, consider him a massive hypocrite with little ground to stand on, and that the narrative vindicating some of his beliefs comes across as absurd. There is little leeway in viewing him. One of the biggest factors behind his divisive nature is his ideology, detailed in Broken Base.
Mitsuki Bakugo is definitely another of the most polarizing figures in the fandom, despite only having one significant appearance in the aftermath of the Hideout Raid Arc. There's lots of debate over whether she's an abusive bully or simply a take-no-shit parent:
Detractors point out that in Mitsuki's one scene so far she smacks Katsuki for literally no reason while not only showing No Sympathy over her son being kidnapped, but seemingly blaming him for the incident. They cite Bakugo blaming himself for All Might's retirement as being Mitsuki's fault. In particular, Mitsuki-haters emphasize that she essentially punishes her kid for turning into aGeneration Xeroxof herself. They also tend to accuse fans of glorifying child abuse towards people they don't like. Not helping is Bakugo's claim during the remedial course that using force to keep people in line is "how (he) was raised."
Fans usually sympathize with the fact that a kid like Katsuki would be a living nightmare to raise, especially considering how Weak-Willed Masaru is. They highlight that Horikoshi clearly intends the behavior to be Played for Laughs and that basically none of the other adult characters have gotten Katsuki to change his behavior by themselves. They also accuse detractors of jumping on any justification for not holding Katsuki responsible for the arrogant cruelty he shows. A few viewers also notice that Katsuki's behavior in the scene in which he and his mother talk to Aizawa, particularly continuing to talk back to her even after she smacks him, is not consistent with that of a traumatized abuse victim, hence why they don't think her treatment is meant to be seen as abusive.
Some dislike Eri because they believe that she serves as more as a plot device than an actual character, while others like her for being an adorableWoobie. The other issue is that her power is essentially able to undo anything, meaning that she endangers the plot's drama by showing a capacity of undoing All Might's injuries and Mirio's lost Quirk. Her fans, however, point out that this has been already taken care of in the sense that she was shown to not be able to use it properly and it is dangerous to train her to even use it, considering her power has to affect living things.