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.
Mizuki, partially because she is seen as kind of bland and her departure to Dream Academy for rather poorly written reasons.
The Dream Academy girls are well accepted among the casual crowd, but in serious circles they are hated.
The fact that two of them use already established brands instead of new ones is also a point of contention.
Sora and Maria. While they do have their fans, they are disliked for being a Mary Sue and The Generic Guy respectively.
Otome's whole schtick is seen as either cute or idiotic.
The Angely Sugar, Spicy Ageha, and Love Moonrise brands are becoming this due to their overexposure.
Akari, due to fears of her replacing Ichigo somehow which is confirmed to happen in the upcoming 3rd season.
Broken Base: Fans are divided with the second season. Some welcome the rival school Dream Academy, its characters who are new, and the competition between the two academies. Others believe that S2 would be better off without them since they do not give room for the less established fan-favorites from the previous season (Shion, Hikari, Sakura, among others). The fact that Mizuki transferred to Dream Academy gives her haters another reason to loathe her.
Canon Sue: It's very hard to see Ichigo as anything else but this, mainly because in spite of having tons of characters, the anime focuses more on her and her personal triumphs, even against obstacles she really really should not succeed.
On the other hand Ichigo is the central character and Audience Surrogate when it comes to the workings of the show so it makes sense for her to get the most focus. Ichigo also doesn't win everything such as the third spot in Tri-Star but when she does she goes through very intense training even for fairly simple obstacles.
Fan-Preferred Couple: While Ichigo/Aoi is no surprise, Ran/Yurika is the more obvious and popular. In Season 2 however, fans jumped ship to Kaede/Yurika after their episode together.
Fanon Discontinuity: Many fans dislike episode 13 due to the reveal that Aikatsu Cards only fit one body type, and a very thin one at that..
Some fans are treating the entire second season as this.
Franchise Original Sin: The first 3 episodes of second season is criticized for reusing the same insert song, which is a rock remix of the second insert song of the first season. This also happens in the first season, albeit that it was the first two songs that get recycled for 3 episodes as well.
Rescued from the Scrappy Heap: Mizuki managed to win some respect from her critics after they watched her background story near the end of season one. She even got more love in season two when she returned with a new partner and an agenda of her own.
Rooting for the Empire: Non-villain example. Many fans wanted WM (Mizuki and Mikuru) to win the Twinkle Cup instead of 2wingS (Ichigo and Seira). The former group lost.
The Scrappy: Sora and Maria are often seen as this, mostly due to being seen as "mary sues" as well as their kind of interchangeable personalities.
The entire Dream Academy can be counted as well for being too much of a friendly rival to Star Academy.
They Wasted a Perfectly Good Character: One common criticism for the anime. Some fan-favorites do not get a fair amount of screentime; such as Hikari, who is supposed to be The Rival for Ran but only shows up in a few episodes; and Shion, who is Aoi's partner in their detective show, remains in the supporting role, although she has a couple of episodes dedicated to her.
Akari's roommate from season 2, Yuu Hattori, never really stood a chance. She had some appearances in Season 2, but gets sidelined at the very beginning of Season 3, thanks to a terribly convenient roof leak. Her cool-idol type essentially got given to Sumire while her short blonde hair got passed to Hinaki.
Visual Effects of Awesome: The CG concerts of the anime's second season are very good looking. So does the 2014 update to the arcade game. Strangely enough, the latest game, Aikatsu 365 Idol Days for Nintendo 3DS, does not use the improved CGI the anime uses.