Taki-sensei's teaching methods have two prominent interpretations among the fans: either he subtly manipulated the club to choose nationals so he could pursue his own ambitions and justify his instruction methods by saying it was their choice, or he's really just doing what he was told to do while balancing politeness and honest harshness to actually get the club to work properly. Similarly, there's a lot of debate on whether or not his methods are effective, plain Jerkassery, or an odd combination of both.
And then episode 9 reveals that Taki-sensei is apparently well-known among the musically-inclined in Kumiko's hometown. Even Kumiko's older sister mentions that she predicts an upsurge in band-interested students enrolling in Kitauji simply because Taki-sensei is there. This sheds a whole new light to his teaching methods.
Broken Base: The nature of Kumiko's confession to Asuka on whatever it's romantic, or just simply platonic is debated heavily by fans, with convincing arguments from both sides.
Contested Sequel: Compared to the warm reception of season 1, opinions on season 2 were more divided. The main problem most people have is that the season feels rushed and cluttered; it combines two novels' worth of stories where the original only worked with one, so the series abruptly shifts in tone after finishing its first arc. It also forces a lot of major characters from the first season into background roles, and reduces Reina's characterization to an independent and headstrong young woman to focusing only on her briefly-mentioned crush on Taki-sensei. From Kumiko/Reina shippers, it also receives heat for shifting the focus to Kumiko's relationship with Asuka, and doing so in a much clumsier manner than Kumiko and Reina's interactions in the first season—see Strangled by the Red String below. However, it's generally agreed that everything besides the story was on par with the original series, especially the animation.
Given the increased focus on Kumiko and Reina's friendship in the anime, especially the romantic side, it should be no surprise that Shuichi, whose romantic interest in Kumiko is reciprocated in the books, has become quite the hated character.
Taki-sensei gets this a bit too but because Reina's interest in him is dropped as soon as it was brought upnote At least in Season 1; it becomes more prominent in the second season. Fan hatred seems more oriented to how he was used just to prove Reina isn't gay then toward Taki himself.
Vice President Asuka is getting a lot of support from the fans, thanks to her cheerful personality and her seiyuu. To the point that she is arguably the most popular person of the show. Tellingly, she's pretty much the Deuteragonist of Season 2, her subplots, Hidden Depths, and dynamic with Kumiko receiving even more focus than in the books.
As the anime's season advanced, the other major third-year students also gained a lot of popularity.
Natsuki was fairly forgettable at first, but now has quite a fanbase.
Fanon: You'll be hard pressed to find a fanwork that doesn't depict Kumiko as a lesbian. Even many fans consider her book incarnation to be gay, or at least bisexual with a heavy lean towards girls, due to the fact she's so oblivious to her feelings for Shuuichi that she needs others to tell her of them.
Needless to say that thanks to a lot of teasing (or baiting), Kumiko x Reina is the most supported ship.
Yuko and Natsuki has become popular due to the two's Vitriolic Best Buds style relationship. It's a popular Ship Mate to any couple, especially Kumiko/Reina.
For those of the fans that actually bother to pair up anime!Shuichi with anyone, he's usually coupled with Hazuki. It helps that he has a bit of Ship Tease with her and she has a crush on him. Other possible candidates are his best friend and even Taki-sensei, since he's chummy with both of them.
Kumiko x Asuka is starting to get support following Kumiko's confession to Asuka in the finale.
Fountain of Memes: Kumiko. Pretty much anything she does, up to and including the small nuanced noises she makes in every episode ever, the very expressive faces she makes every now and then, or just some of her awkward weirdness in general, is turned into a meme by the fandom.
He Really Can Act: Many people have praised Tomoyo Kurosawa's acting for Kumiko's complex character. Things from the nervous stuttering to the nuances in how she says single-word lines have been commended.
Heartwarming in Hindsight: The Kumiko x Reina shipping started before the anime began and is already popular on its own. However, it got so much support before the anime ended when the US Supreme Court declared same-sex marriage to be legalized in all 50 states. Some sites like Crunchyroll would go on to support the fan-favorite pairing.
Memetic Mutation: Kumiko noises Explanation Thanks to Tomoyo Kurosawa's performance of Kumiko's awkward character, the obscure noises Kumiko makes with her mouth in every episode ever has become a hit among the fandom, to the point where there are several YouTube videos dedicated to it, calling it "God's work".
Moe: Almost every single one of the girls can account for this in some way or shape. You can just pick your favorite out of the cast.
Relationship Writing Fumble: The way the relationship between Reina and Kumiko is handled in the anime caused a lot of viewers to expect a romantic payoff. The novels fare a bit better, even though a lot of fans think that Kumiko's decision to date Shuichi doesn't make much sense either, considering the utter lack of chemistry between them.
Rescued from the Scrappy Heap: After Reina retains her solo position for the performance, Yuko got better with the fans thanks to her amusing interactions with Natsuki. The fans would go on to ship them together. The second season portrayed her more sympathetically, further improving her standing.
Second Season Downfall: The general reaction of the Hibike anime's second season is decent to mediocre. They note that the drama seems forced, the first halves of the episodes don't match up in tone with the second halves, and the arc with Asuka is a little spotty. Overall, it doesn't match the quality the first season brought to the table.
Anime!Shuichi gets this label from some anime watchers. Because his role was reduced to a minor supporting character, thus leaving out any opportunity for significant character development, his cruel rebuttal of Kumiko's offer to spend Graduation Night with her family has earned him some fans' loathing. It doesn't help that he barely shows remorse for having done this.
The majority Kumiko/Reina fans versus the minority Kumiko/Shuichi fans.
The final episode of the second season caused this due to Kumiko's ambiguous confession to Asuka. Kumiko/Reina versus Kumiko/Asuka is in full effect. Prior to the ending, the latter ship was rarely even taken seriously.
Since the anime's Shuichi is largely Demoted to Extra, the Love Triangle plot becomes this due to the elimination of much of its buildup and foreshadowing.
Kumiko confessing to Asuka in the finale. Sure, their dynamic was highlighted far more than in the books, but it still feels almost comically abrupt, with her quite literally literally interrupting what was being set up as Shuichi's resolution. Even Asuka seems mildly confused before brushing it off in her usual wishy-washy manner.
They Wasted a Perfectly Good Plot: Season 2's big scene at Nationals isn't shown at all in Episode 12. The audience never gets to learn of the fruits of all their hard work this turn around.
While Reina's need to stand out and go her own way resonates with many Western watchers (who normally come from cultures that value individuality and personal initiative), her attitude is actually considered disruptive in-story as well as among Japanese viewers, due to the communal and hierarchial nature of Japanese society.
The social hierarchy among the students is key to a great deal of the club politics and drama. Clubs around the world all have their politics (and Japan is hardly unique in expecting younger students to defer to the older ones), but as The Incident that took place last year is based around the slacker third-years bullying and harassing the younger members who wanted to play properly, you might expect that they would be disciplined for it or told to leave the club. Instead, nothing is done, most second years quit out of frustration and, rather than being angry at the bullies, the remaining then-second (now third) years seem to blame themselves for not shutting up and tolerating it until the bullies graduated. Upon hearing the story, Kumiko is more scandalized that Natsuki openly challenged them and called her seniors a bunch of divas than that the older students ganged up against the younger ones. Apparently, no teacher saw fit to step in and protect the younger students, even when they ended up with a pile of resignations from the club. The seniority system is also the reason there's so much debate over who gets solos and who gets into the competition team — some members assumed that third years would get automatic entry, then the second years, and the first years would be picked last, if there were any spaces left. When their new teacher holds auditions, many second and third year students are shocked.