Shouma: Normal Bishonen nice guy who is trying to save his possibly doomed sister and help another girl at the same time, or Cute PsychoStepford Smiler? If it's the latter, why? Has he been that way since the beginning and just needs an excuse, or is it out of sheer force of trauma and guilt after the parents had to run away and the siblings were left on their own?
Kanba: Just how far will he go for Himari's sake? And is he in love either with Himari or with the Princess, specially after episode 12?
Himari: Is she truly that oblivious to what happens around her, or does she know more than she lets on (and may be in denial due to her grief)? Also, when she gets possessed by the Princess, how much of her conscience is left: does the "real Himari" fully retreat, or can she influence the Princess's thoughts and behavior (and/or viceversa)?
Tabuki: Is he really that much of a Cloud Cuckoo Lander? And it he's not, then just how much is he aware of, and what secrets he keeps in his own mind?
Masako: Like Kanba, how far will she go for Mario? And what did Kanba exactly do to her, anyway?
Sanetoshi: What are his true plans, and what risks will he be willing to take to meet his fated one again?
Angst? What Angst?: Ringo shows absolutely no signs of trauma from being kidnapped, drugged, and raped by Yuri. The closest she gets is a small amount of hesitation when Yuri calls her to say that last time was "fun" and then invite her to dinner.
Ass Pull: Some fans believe Yuri's relationship and backstory with Momoka to be this, seeing as it comes with little foreshadowing. It doesn't help that the reveal comes hand-in-hand with a turn of events that come off as AssPulls themselves (see They Wasted a Perfectly Good Plot and Double Standard: Rape, Female on Female for more details), and that the events make the reveal itself look even more like an Ass Pull given the absolute lack of attention given to Yuri's attraction to Ringo both before and after. Not to mention that the event is resolved shoddily and heavily smacks of the writers writing themselves into a corner and hastily retconning themselves out of it, probably because they couldn't figure out how to fit Yuri's plan to make Ringo into Momoka into the plot or handle the consequences of Ringo being raped, or both.
Special mention goes to Shoma's attempt to rescue Ringo from being raped by Yuri. It's hideously contrived to the point of absurdity and doesn't even manage to accomplish anything, both short- and long-term.
Depending on where you go, you either think that Ringo is an adorable and funny Plucky Girl who knows what she wants and will do her best to get her "Destiny" fulfilled and deserves a lot of sympathy for her genuinely sad backstory, or that she's a stupid, delusional, psychotic, stalker who's also an attempted rapist who totally deserved to have Himari's possessor insulting her all over as well as being locked up in a ward.
And others have taken a third option via feeling bad for both of them, since it's not exactly an easy situation for either kid. Then again, it's noticeable that ever since episode 6 explained her backstory, the sympathy for Ringo has steadily increased, reaching its peak in episode 14; people who professed their hate for her at first have been spotted rooting for her now.
While Shouma still has many fans, he has also had detractors referring to him as "weak" and "useless" and "inferior toKanba". Then episode 19 rolls around and shows that Shouma was the one who took Himari out of the Child Broiler and into the Takakura family. Cue the rage of many Kanba fans on /a/.
Double Standard: Rape, Female on Female: Penguindrum has the adult Yuri's attempted rape of Ringo, an underage sixteen-year old girl. On one hand, the rape isn't exactly portrayed as okay, and it's made clear that the rapist is a messed-up person. However, the rape is used to develop the rapist's character and segue into the rapist's backstory, which gives off the vibe of rationalizing and excusing her actions, whereas the victim is reduced to a mere prop and has an uncomfortable amount of Male Gaze fanservice applied to her — which may have been intended as Fan Disservice, but the sheer exploitative nature of it makes the claim fall flat. The details of the rape are inconsistent from one episode to the next, with the rape being portrayed as rape one episode and as a titillating seduction in the next, and whether or not the rapist actually went through with it or not is ambiguous; the rescue attempt is played for comedy, and only happens due to a hideous amount of contrivity. Furthermore, the victim is completely unaffected by the ordeal and the rapist gets off with absolutely no repercussions for her actions, and the whole scene affects practically nothing in the grand scheme of the story and is quickly forgotten afterwards, save for being the subject of a brief joke a couple of episodes later, wasting a lot of character and story potential on top of everything else; in fact, the scene could be completely cut out of the story with little to no loss. The shoddy handling of the scene really smacks of the writers writing themselves into a corner and hastily retconning the story in an attempt to fix it, and given all the above, it's no surprise that some viewers consider this part of Penguindrum to be one of the lowest points in the series.
Esoteric Happy Ending: Sanetoshi warns Kanba that he and Shouma will be erased from existence and leave not so much as a grain of dust behind. This is (mostly) true, but unlike Sanetoshi, the brothers look upon this as a good thing.
Freud Was Right: During the Seizon Senryaku sequence, something long and pink shoots from the base of the Princess of the Crystal's bear into Shouma and Kanba's.
"Funny Aneurysm" Moment: The ED becomes a lot harder to watch after Episode 9 reveals that the Double H girls used to be Himari's friends and that the ED is a wistful reminder of something that can't happen for her.
Also, seeing the beautiful Yuri in a princess-like pose and garb in the first OP is harsher when you learn how her father wanted "to make her beautiful"...
When the Princess of the Crystal refers to Shouma and Kanba as "lowlifes that never will accomplish to anything", it's really freaking funny. When little Himari refers to herself as "someone who never accomplished to anything" as she's in the Child Broiler, it's really freaking sad.
Seeing Kanba blush when he's being complimented by his parents in episode 19 is cute, but when you find out he's actually talking to his parents' rotting corpses and the living parents are just a delusion he constructed to not feel horrible about what he's doing it's just heartbreaking.
Another candidate is Sanetoshi Watase.Specially after episode 13. And after episode 23, if you don't think he irrevocably crossed the MEH
For some, Kenzan Takakura may have had some inklings of this as well.
Memetic Badass: Kanba is already badass enough in-story, but his 4chan fans take his badassery even further. Because this guy will punch gods and goddesses and kick whatever/whoever is on his way FOR HIMARI'S SAKE!
Mind Game Ship: Sanetoshi with anyone. Examples are Himari, the Princess of the Crystal and speciallyKanba.
Moral Event Horizon: Sanetoshi had done plenty of shit before episode 23, but he seriously crossed the MEH not just by being the one who "killed" Momoka via transforming her in the two Penguin Hats when she tried to stop him, but by using Masako's "death" and his revival of her to push Kanba further into his plans, the setting up Ringo to be seriously injured and mocking her as she was burning to almost death. All to destroy the Destiny Diary.
Some people believe that the Face Heel Turned Kanba also crossed the line by becoming the KIGA leader and orchestrating the Teddydrum bombingsand by participating in Sanetoshi's almost murder of Ringo to destroy the Diary.
Narm Charm: The fight between Masako and Yuri in Episode 15 is so ridiculously over-the-top (it involves Yuri swinging ping-pong paddles to swat Masako's slingshot balls, for crying out loud!) that it warps back to ridiculously awesome.
The whole scene where Ringo remembers her parents eventually deciding to split up is reenacted via stuffed animals. It's ridiculous, but considering she was a young child at the time it actually works and conveys how traumatic it was for her.
By the same token, most of Ringo's Imagine Spots are so over the top ridiculous that you can't help but laugh, but they also managed to convey just how screwed up the girl is.
Pacing Problems: While it did flesh out her character, as well as introduce other characters crucial to the story, one may wonder if it was really necessary to spend seven consecutive episodes on Ringo and her arc.
Ship-to-Ship Combat: As of EP 19, there has been serious ship wars between Kanba/Himari, Shouma/Ringo and Shouma/Himari.
Strangled by the Red String: Some complains have been raised in regards to the relative lack of development between Momoka and Tabuki's bond, specially when compared with how Yuri and Momoka got almost whole episode to explain their friendship while Momoka and Tabuki's own took less than 1/3 of another.
However, this seems for the moment justified — Tabuki says he doesn't understand Momoka's actions either, and he is an Unreliable Narrator as well.
They Wasted a Perfectly Good Plot: Yuri trying to make Ringo into Momoka. Introduced in episode 14, then shoved into the background by other plot developments and pretty much dropped a few episodes later with little fanfare; not to mention the non-existence of the consequences of Yuri's rape, as well as the incredibly awful handling of the event itself, wasting a lot of potential for character development. It doesn't help that the whole situation feels shoved into the plot for drama and to quickly explain Yuri's involvement in the plot to begin with, with said explanation clashing with earlier plot developments and characterization.
What Measure Is a Non-Badass?: Shouma gets potshots (ranging from mere affectionate jokes to straight-up bashing) for not being as action-geared as Kanba. Even more so after episode 19.
There's also the fact that both of the canonically-confirmed lesbian characters in the show are a rapist, attempted or otherwise, who gets away with her crimes, and an overly-jealous ex who stabs someone and gets away with her crimes, respectively; made more insulting by the fact that the latter is a minor character with little-to-no impact on the plot that could've been easily cut out, and the former ultimately ends up with a male character anyway. Pointed out here and here.
Unintentionally Unsympathetic: Yuri is definitely played up to be sympathetic in the narrative, but it's hard to regard her as such when she unapologetically kidnaps, drugs, and forces herself on an underage teenage girl out of hideously selfish reasons, gets away with no repercussions, and even makes a second attempt which luckily doesn't get off the ground. Not to mention that at the same time, Yuri also plots to murder a sickly, completely innocent child just because of her family relation.
Ringo, once her Freudian Excuse is revealed. Even more so after episode 14, in which the closest to a best friend/love interest she has pushes her away due to his issues, and then she's pretty much kidnapped and raped by Yuri.
And after episode 9, Himari. Then episodes 19 and 20 kick in and turns this Up to Eleven.
Woolseyism: Discussed by the (fan) translator himself here. In particular, the "Lol wut? U mad?" line is a translation of some outdated slang that Sanetoshi uses. Since Sanetoshi is also quite a troll, it also makes sense for him to use it. Most prominent in the opening and closing songs:
The GG subs use "Everybody went nutso!" instead of the more literal "Everyone's gone crazy."
"Now remember the spectacular past" vs "Now remember the time when we shone."
"From the other side of the binary world line," vs "From the end boundary of a world made of 0's and 1's."
Yoko Oh No: Three girls who were dumped by Kanba (Asami, Yui and Chizuru) form a club named "Heart-Broken Victims of Takakura Kanba Assocation". As such, Yui and Chizuru corner Kanba at some point, asking him if he has a new girlfriend or a girl has taken their places in his heart. Then, local Action Girl Masako then shows up and wipes their memories away... just like she did to an hospitalized Asami right beforehand.
Averted in the case of the in-story famous Takarazuka actress Yuri Tokikago, who is dating Kanba and Shoma's teacher Keiju Tabuki and hasn't had problems coming from fans as far as the story shows. This doesn't change when they get engaged, Yuri announces that she will retire soon from acting and Tabuki moves into her Big Fancy House. The only one really objecting is Ringo... and well, this is Ringo we're talking about.