Did Tamaki create the Host Club merely for the fun of it and in order to show off his good looks, or did he do it out of loneliness due to his background in order to gather "family members" he himself fancied, with the excuse of wanting to entertain ladies? Or maybe he did it for the others? He looked at the states of his colleague's lives (the family-pressured Kyoya, the antisocial twins, etc.) and decided that they needed the "club family atmosphere" more than he did.
Haruhi falling for Tamaki. Did it symbolize her Character Development into a more caring, outgoing person, learning to love Tamaki for who he was rather than just be annoyed at his quirks, or was she regressed into a stereotypical shoujo heroine whose only purpose is to fall for the main male lead and blush like a maniac around him, taking away her indifferent streak that made her so interesting in the first place? The believers of the latter seems to be a minority however, and Haruhi did more or less return to her old self after calming down from her realization, keeping her snarkiness and ability to handle Tamaki just fine next to having romantic feelings.
Did Kyoya genuinely try to rape Haruhi, but was talked out of it, or was it just a test like Haruhi theorized?
There are some fans (mostly a certain type of fan) who believe that Renge might at least subconsciously have known Haruhi's secret.
Did Hikaru actually like/love Haruhi? Or was he just emotionally attached to the one person that could tell him apart from his brother? It's a bit more on the former side in the manga.
Hikaru appears to get over his feelings for Haruhi really fast, going from nearly crying at the knowledge that he lost, to having no qualms about secretly following her and Tamaki on their date along with the rest of the club just a few days later. Though it might have been to symbolize how much he had grown and thus were capable of keeping his emotions in check, or he had been processing in getting over his feelings for a while.
In-universe: when Haruhi finds out about Tamaki's past she's impressed with how one wouldn't be able to tell by his attitude.
Renge, who is either really funny or really annoying depending on who you talk to.
Haruhi. Fans will either find her apathy and snarkiness around shoujo males whom the heroine usually blushes around 24/7 to be refreshing and serving as a better "role model" for the average teenage viewer/reader, or make her bland and contributing little to the plot other than to be a generic snarker, and find her being an oblivious Dude Magnet for no other apparent reason but that she's "cute!!!" to be obnoxious. It also got turned around later in the manga when she realized that she'd fallen for Tamaki; fans who didn't find her smitten attitude around him to be hilarious believed that she had turned into exactly the stereotypical heroine they had loved her for not being. Even as she more or less returned to her old self after calming down from her initial reaction, it still didn't keep the fans who had expected her to be a unique heroine who didn't have to end up with the male protagonist from leaving and never coming back.
Tamaki is debatably the most popular character in the series, but there are also those who just find him annoying.
Honey. His way of acting makes him either beyond adorable or seriously creepy.
In one episode there's a scene where the twins are lying in bed in front of their maids. One of them pulls off the covers and reveals he's wearing a colorful elephant head on his crotch, that blasts confetti into the air. This is not given any lines, and it goes onto the next scene quickly afterwards, with this never being explained or mentioned again, except for a brief moment that implies that they plan on doing it again. (The elephant head is a common way to censor out nudity, so that's probably what that was.)
The twins being randomly dressed as girls at a party during their childhood, though only in the anime. (In the manga, it's established that their mother had designed a children's clothes line at the time, for which she used the twins as models.)
Bizarro Episode: There was the random episode that was a parody of Alice's Adventures in Wonderland featuring random single-episode characters from throughout the first season showing up again... In the anime, the end is a Tear Jerker: the whole thing is a dream of Haruhi's, and the one playing the Queen of Hearts is actually her Missing Mom Kotoko.
Haruhi's father's friend Misuzu is extremely popular, the few times he does appear.
Girl Show Ghetto: Averted. Despite being aimed at a female audience, many guys enjoyed watching the show, mostly because of the light-hearted humor and the constant self-parody of Shoujo clichés.
Growing the Beard: The series starts out as mostly a comedy and (as mentioned above) Affectionate Parody of the standard shoujo clichès, with the occasional yet brief serious or heartwarming moments, but becomes deeper and more serious the longer it goes on, revealing hidden depths within many of the Host Club members along with other characters such as their families, and testing of how strong their bonds truly are (specially in the manga, though some of it in the anime too).
Harsher in Hindsight: An in-universe example: in Episode 14 of the anime, Akira Komatsuzawa, the newspaper president, becomes obsessed with the thought that Tamaki can't be such a perfect Nice Guy as he appears to be, and there must be something evil or scandal-like to him, or even if there isn't, he'll make something up. In the end he's thwarted by the rest of the Host Club, and Haruhi's left with her doubts that there would be anything bad to find about Tamaki. Come the last two episodes of the anime (or Chapters 25-26 of the manga) she (and the audience) learns how Tamaki does have a sad and scandal-worthy past. The short version is that he's a mistress' child, and he's forbidden from seeing his mother ever again by his Evil Matriarch grandmother. Towards the end of the manga however, it turns out that their customers knew about this, due to always wanting to check out the background of their favorite hosts, and because of Tamaki's kind and cheerful self they were fine with it.
Tamaki's scolding of Haruhi for trying to protect customers from a pair of drunkards, which seemed to boil down to "Girls are too weak to fight boys!" rather than "A small person shouldn't take on two brutes." Although, the Host Club points out that their main concern was her attempt at taking them on alone when she could have called for help, which needlessly put her in danger, and she really had no fighting skills to begin with anyway.
There's Kyoya's famous threat of rape of Haruhi later the same episode, which is quickly shrugged off by the characters and narrative in whole, being treated as acceptable because he was "pretending in order to teach her a lesson" and Haruhi, at the very least, doesn't believe he has any intention of following through.
Western fans have a harder time swallowing the obsession with characters like Honey or Shiro, whose admirers specifically like that they appear to be, or actually are, a pre-pubescent child and act like one.