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.
Angst? What Angst?: Hayate, though he does finally crack in Season 1 when he apparently failed his Entrance Exam to Hakuō Academy, bitterly weeping in resignation that "such a good school is too good for someone like me anyway." One of the rare Tearjerker scenes in the franchise played completely straight.
Base Breaker: Athena. Some people love her, thinks the story is better because of her and Hayate's one and only and are saddened by her leaving in chapter 265. Others think the story was doing fine without her, or even go to the extreme of slutshaming her for having sexier looks than the other girls. Part of this is probably because of Hinagiku shippers.
A character poll in chapter 110 of the manga put Hinagiku's popularity at number one; her number of votes is about twice the number of second placer Maria and just a few dozen votes short of Maria, Nagi and Hayate's combined. Looks like Nagi is just a tad bitter now...
The first ED for the second season is completely dedicated to Hinagiku. It's a love song to Hayate sung by Hina in the framing device of a karaoke bar, complete with plenty of blushing and a flood of multicolored hearts. Aside from Those Three Girls making periodic singsong remarks, no one else is in it. It's a nice Ship Tease for Hina fans but feels somewhat bizarrely out of place in the grand scheme of things...
Given other material, the song doesn't seem to have anything to do with Hayate at all (instead focusing on the friendships). The Hayate-themed ending was probably just Hinagiku's mind playing with her, since that was what the arc was about in it's entirety.
Fan-Preferred Couple: The show is already rather unusual since not only is there no Official Couple yet, but the chance of the male and female leads never getting together is seriously, legitimately, a possibility despite occasional Ship Tease (but everyone gets that in this show).
Hinagiku and Athena both qualify, each with their own very vocal group which even separately seem to dominate the forums. Seriously if you didn't read the series you'd think that they were the main heroines instead of Nagi and Maria. This is especially evident with the results of the 3rd Character Popularity Poll: Hinagiku retains 1st with over 4000 votes, but Maria and Athena switched places and Athena is 2nd with 2100 votes actually being the first to get more than half of Hina's votes which is a serious accomplishment given Hina's widespread popularity even outside the show's fanbase. Naturally, these two seem to get the most Ship Tease, though Hata being Hata everyone gets their fair share.
Possibly a cause for Maria to be getting pushed into the background, while Hinagiku seems to be pushed into the vacancy
Magnificent Bastard: Opposing Mikado (though in-story she doesn't even seem aware of him yet), Miki Hanabishi is proudly known for her plots to make trouble for Hinagiku, in fact it's clearly stated in her profile that she enjoys it, but generally seems to be using such tactics to push Hina to the forefront of the competition for Hayate'saffection. While at the same time pursuing Hinagiku with her own romantic interest.
There ought to be a law against anyone being as moe as Nagi's mother was as a child, 30 years ago.
Moral Event Horizon: Hayate's parents cross it in the first chapter, when they try to sell his organs to "those very nice people". It's later revealed throughout the course of the series that their depravity knows no bounds, culminating in the revelation that they pawned off the ring Athena trusted him with. It's absolutely no wonder that when asked whether he'd return to his parents if they ever had a change of heart, Hayate himself said that "there are no second chances". After what they put him through, can you blame him?
Athena's "lesson" on Hayate (which involved swords and wasn't fun) after neglecting to tell her about his kiss with Izumi when he left the castle, which ultimately resulted in the latter's compulsive habit to be stupidly honest overriding his sense of self preservation, often saying the wrong thing at the right moment to get him brutally assaulted.
Ship Sinking: When 'Alice' enters the scene, she recognizes Hayate and knows possibly Hinagiku has the sword she's looking for, but there's no indications of the affection she confessed to at the end of her arc. In fact, she actively denies any relationship with Hayate during her chapter.
Even Kotetsu would have the emotions carried over if he lost his memories.
The Woobie: Hayate, both to the viewers and to multiple characters in the story. Usually hearing about his parents is enough to make most characters instantly want to hug him and make it all better.
Hinagiku is getting to this level within the manga currently
Toy Ship: Hayate and Athena when they were around 6.
Isumi. Look up "hug me" in a dictionary and her picture would probably be there.
Hinagiku later on the manga qualifies as well. In addition to the emotional wringer she was put through in Greece, the recurring anxiety and occasional depression over "being stuck on the comedy end of Love-Comedy" since she realized her feelings for Hayate, and her many forced confrontations with her fear of heights, she's also become the butt of more jokes at her expense by her friends and the plot itself as opposed to the awe and respect she got at the start of the series. Oddly enough, this could actually be Character Development since she was more aloof in the beginning and out of touch with her true feelings - as she's opened up and showed her vulnerable side to more characters, it gets poked fun of more and more. It doesn't help that several characters, particularly Miki, Chiharu, and Aisa, tease Hinagiku and put her in these situations on purpose to provoke her flustered Tsundere outbursts.