Author's Saving Throw: A common complaint from fans, especially from fans of the original visual novels, is the lack of touching on for the boys' backgrounds in the anime which gave a lot of characterization to them. Season 4 has now shown hints and elements of the backgrounds of Masato, Natsuki and Otoya.
Another common complaint with the fans, this time from both fans of the original visual novels and games, is how none of the boys have had the guts to not just confess to Haruka but ask her for a clear answer in the anime. Come Season 4, in enters Van Kiryuin who not only confesses to Haruka but asks for an answer when he asks her out. Bonus points were given to both Haruka and Van for taking the confession and rejection respectively in a surprisingly calm and non-cliche manner. This was part of why Van became quite well-liked.
Awesome Music: The music is composed by Elements Garden, and some of the tracks definitely apply, like "CRYSTAL TIME".
The duets between STARISH and HEAVENS members definitely qualify. Prominent examples being "Lasting Oneness" (Masato and Kira) and "Mighty Aura" (Tokiya and Eiji).
Big-Lipped Alligator Moment: Cecil's appearance in the first season. Only Haruka meets him, and he's never mentioned again until he comes back to join the Master Course in the second.
Also happens in the main game's routes (except his own): he appears suddenly to give some advice to Haruka, and later dissapears for the rest of the route with no explanation.
The second season's ending. Everyone floating in the sky and reaching space.
Broken Base: For some fans of the original game the anime dumbifies the characters, takes away their characterizations and backgrounds and loses the spirit of the original. However, there are also many fans who knew the franchise thanks to the animated series, and who prefer watching the guys interacting with each other rather than engaging in romance with Haruka.
The use of CG in "Maji Love Revolutions". Some find it cheap, others don't mind and consider it makes the song different to the previous endings.
The focus of the anime being given to HEAVENS.
Die for Our Ship: Haruka is a girl in an otome game. Three guesses as to what the rabid yaoi fans do to her.
From Season 4, Tokiya/Eiji. Their episode seriously can be seen as a typical shoujo story where Eiji is seen as an angel by Tokiya and Haruka, and with Eiji even giving Tokiya his number.
The whole episode practically involves Tokiya feeling conflicted between staying with STARISH or taking Eiji's offer and joining HEAVENS after seeing how hard the latter works and takes things seriously. But easily, one can interpret it as Tokiya getting interested in HEAVENS because of Eiji.
Launcher of a Thousand Ships: Haruka. Considering she's the heroine of an otome game, she already has at least six good-looking guys after her; she can be paired with Otoya, Tokiya, Masato, Ren, Syo, Natsuki, Cecil, Hyuuga, Ringo, Reiji, Ranmaru, Camus, Ai... and the list goes on.
Of the boys, Syo has one of the largest shipping pools, nearly rivaling Nanami; he gets shipped with Natsuki, Satsuki, Ai, Nanami, Kaoru, Cecil, Ren, and Nagi. And those are only the regularly recurring ones.
Les Yay: Haruka gets REALLY happy when Tomochika visits her.
Most Annoying Sound: season 3, "revolution". You'll hear it three times too many per episode, if you're lucky.
This is even lampshaded by Junichi Suwabe (Ren's voice actor) who made a tweet saying they "over did it" with using "revolution" in the final scene of the first episode when it first aired.
Eiichi Ootori was hated for being really creepy towards Haruka along with his Slasher Smile. However, he garnered sympathy and more fans when it was shown that he was like a big brother figure to the rest of his band, and legitimately just wants what's the best for them.
As of the end of Season 4, all of HEAVENS' controversial members have been upgraded to this due to the highlight the better sides of their personalities got. They are still a Broken Base case for some parts of the fandom, but in general they've been well-accepted.
The Scrappy: Haruka, for fans of the anime (mainly disliked for her generic personality) and Cecil for both game and anime fandoms.
Also from HEAVENS, Nagi Mikado is highly disliked by a lot of fans due to his status as being The Fake Cutie and how much of a Jerkass he is with his debut showing him slapping Natsuki's hand away and told him how dare he think they can shake hands. A lot of people really hated the kid in his episode in Season 4 with Natsuki where he taunted Satsuki about Natsuki's Multiple Personality disorder and was even considering telling the news about it.