These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Arc Fatigue: The fighting-manga arc has been going on pretty much nonstop since the series came close to being canceled (see Genre Shift below). They stopped for a single chapter, if only for exposition to introduce the new set of villains, then jumped right back in.
Ass Pull: Lots. Especially with combat — not that it was handled badly. That is, until Naze Youka being Medaka's never-mentioned-before older sister Kujira. There are hints (her eyes being the same as Medaka's and Maguro's being the main one) but only subtle ones, and it still feels like a huge stretch.
The 15 story research lab under the school (re-purposed as a very big gym). Which no one knew about, supposedly. The Genre Shift has not been kind to this series, although we may be seeing Writer Revolt.
Black Hole Sue: Medaka literally contains a wad of blackness so deep it melts Oudo's arm when he tries to take some of her abnormality and so dangerous that he immediately apologizes to her.
Medaka seems to be a deliberate example due to her status as the main character; nearly all plots and characters draw at least some of their motivation from her, whether it be defeating her, admiring her, fearing her, hating her, etc.
Broken Base: Regarding the various Genre-Shifts; some people preferred the slice-of-life elements of the first arcs, some preferred the straight up battle manga, and some prefer the Deconstruction manga.
Ensemble Darkhorse: Aka Aoki. She's been in three chapters, has barely moved from one spot, and didn't even fight. There is already a whole bunch of fanart of her on pixiv. Probably because she's a nurse though...
Kumagawa himself. He starts out as the third Big Bad, and pretty much Medaka's Arch Enemy and complete opposite. His dissonant serenity and overall humor arguably make his character more interesting than Medaka's. And having a status change doesn't deter his status in the slightest.
In the second Shounen Jump popularity poll contest, Kumagawa won with almost four thousand votes. The second place holder (Medaka) did not even reach one thousand votes. And for a while he was either played as The Dreaded or the Big Bad.
His popularity became much higher than even the main character, he's got his own one shot story, and became somewhat of a flagship character for the series. Just compare the amount of fanarts he gets to the rest of the crew on pixiv! Maybe even some people who aren't into Medaka Box know him by now. Much of his success must be related to his Cool Loser and Lovable Sex Maniac traits, also.
Ajimu: If, for example, this world was a manga where you guys were the main characters, if there was a vote for the favorite character, someone like Kumagawa-kun might unexpectedly come out and take first place.
Arguably Youka Naze and Kei Munakata also count as they consistently place in the top 10 on popularity polls despite not being main characters.
More surprising is Nienami Namanie, who has had a ridiculously little effect on the plot compared to her overwhelming popularity in the fandom.
Nekomi is also popular for her Combat Pragmatist ideology while being perky at the same time.
Fan-Preferred Couple: Zenkichi/Emukae, as well as Medaka/Kumagawa definitely seem to be fairly popular, compared to the canonical affection Zenkichi has for Medaka which seems to be reciprocated as of Chapter 140.
The arrival of the "-Abnormals" and "Class -13" — Abnormals that aren't Mary Sue parodies - but rather, Anti-Sue parodies. At this point, all attempts at a realistic tone were completely dropped, in lieu of people who can instantly heal any wound, disintegrate entire buildings, are so awesome the brain literally cannotbearto remember them.
God-Mode Sue: Medaka. Interestingly enough, the story starts deconstructing the concept of a Mary Sue as it goes on. It turns out she's very good at mimicking other people's abnormalities, minus the downsides. The series was largely using the entire concept for laughs even before the shift, as Medaka was often shown to be utterly clueless and ridiculously naive for all her intelligence and physical prowess, with her boundless sincerity actually more effective than her other abilities.
Sort of inverted in that the entirety of the -13 class has similar, extremely bad "-Abnormalities", such as the ability to rot anything and everything they touch — including buildings. A character even specifically states the -13 class is basically Medaka's perfect enemy, since she can't even approach some of them without absorbing mental illness or Blessed with Suck, which she would then take Up to Eleven.
Taken Up to Eleven with Ajimu Najimi, who has several quadrillion superpowers. It's as absurd as it sounds, intentionally so.
Then there's Iihiko, who is so powerful he can curbstomp any and all of the above.
Idiot Plot: You would think that in episode eight, where it demonstrates Onigase's strength, the handcuffs wouldn't be too hard for her to brake. At least Medaka had the excuse of 'she didn't think Onigase would want the handcuffs broken.'
I Knew It: That Misogi would join the student council as Vice-President.
Jerkass Woobie: The true nature of the Minus class. To quote one Minus: "Do you know the taste of gas that your parent makes you swallow? How about the taste of licking a perverted teacher's shoe? The taste of sand that your classmates make you swallow? At least you know the taste of a newspaper that was the only food you could obtain? Of course you don't know, because you guys are the happy ones. But those kinds of tastes are just how it is for us, Class Minus 13's members. We have experienced the worst, that's why we can do the worst."
Launcher of a Thousand Ships: Hitoyoshi, arguably. He has Medaka, Shiranui, Kikaijima, Mukae, Ajimu, Naze, Akune, Kumagawa, Maguro, Munakata, and now Tsugiha.
Kumagawa is also shaping up to be this, sharing pretty much all of the above characters, but replacing himself with Hitoyoshi, Maguro with Gagamaru, Munakata with Shibushi, and Tsugiha with Ima. And for some reason there was a picture of him and Shiranui with Hinokage...
Shiranui seems to be an extremely cute girl with an ahoge, but she has flat out said she enjoys following Zenkichi and the crew around because he keeps getting beaten up, and has tricked him into getting mauled by a giant wolf, as just one example. The look on her face when she refuses Medaka's offer of being vice president is pure Nightmare Fuel.
Especially after the Successor arc, she shows no hesitation in wanting to fight Zenkichi without any remorse beforehand. It clearly shows just how unnerving her behavior is.
Ron the Death Eater: Medaka seems to be intensely hated by certain segments of the fan-base, people going so far to say that she is far more evil that Kumagawa or Anshin'in and deserves to die/be raped/be mindraped/treated like garbage/etc. for her crimes of wanting to help people note in a matter of duty rather than true empathy and treat her enemies more like friends, thus isolating herself from those who are her real friends. and not showing Zenkichi favoritism.
The Scrappy: The majority of the fanbase hates Akune for being a direct rival to Zenkichi for being an all around perfect Bishounen.
Rescued from the Scrappy Heap: However, he redeems himself in later chapters as it's revealed it's actually a reformed thug who sees nothing wrong with fighting as dirty as hell (in stark contrast to Zenkichi and Medaka). It doesn't hurt that the romance themes of the series were outright dropped as the series turned into a combat-centric series.
Squick: Taiyaki becomes this when Kumagawa eats it.
Tastes Like Diabetes: Medaka, though rarely. Zenkichi noted it as one of her fortes. You can see rather clearly near the end of Episode 6, when she kisses Kikajima in midair.
They Wasted a Perfectly Good Character: Zenkichi and his family. Earlier on, it's implied that Zenkichi himself has abnormal potential, and this is only given more fire when it's revealed his mother is an abnormal. Yet these points (as well as his overall upbringing) are neglected in favor of making him a Satellite Character to Medaka.
True Art Is Angsty: Naze Youka, who gave up a life of luxury as one of the Kurokami children due to the theory that true genius can only come from living through hell.
IF I'M ONLY GOING TO BE HAPPY, I'M BETTER OFF DEAD
She decides to stop this attitude in exchange for helping her brother save Koga's life.
True Art Is Incomprehensible: Becomes a plot point with Yabaru in episode 7, who wants to capture the beauty of a goddess, but turns down various girls, and even the guys, until Shiranui shows up. The fact that he chose her to represent what he wanted doesn't sit well with the girls.
Uncanny Valley: Misogi crawled straight out of the valley the moment he was first mentioned.
Ajimu Najimi is a parody of the very concept, dialed up (as everything in this series is) to absurd levels.