One of the biggest debates among all fans of anime that's continuously broken the base: Subbing vs. Dubbing. Is it better to watch an anime subbed (in its original language with translation subtitles on-screen), or dubbed (having the voices replaced with the listener's native language)? Some say sub-only, some say dub-only, and others are on a case-by-case basis. The one constant is that there's very little overlap, and very high potential for a flame war any time the subject comes up. "Subs vs. Dubs" is a very hotly contested topic, and one that doesn't appear to be going away any time soon.
The ending the Space Station arc split fans , with some being okay with a temporary Snap Back, while others were unhappy with how rushed this trip was, especially considering that this would have ended up being completely pointless if the discovery about Koro-sensei's new probability to explode (which was reduced to 1%) wasn't brought up later. There was even confusion on whether the series was cancelled in chapter 153, after this statement was brought into the class' discussion.
Those who have read the manga are really frustrated with the anime adaptation for its Adaptation Distillation that completely skipped several A Day in the Limelight chapters for the other students that gave them Character Development, rushing through the arcs, and leaving out several important information that leads to several plot holes in the anime.
Rebuild of Evangelion: Are the films good remakes of the original series, or do take away what made the original series so iconic?
3.33 seems to have caused a divide as well with some people viewing it as a cop out from the ending of 2.22 and some changes made to several characters actions being unexplained.
Sailor Moon: The original English dubs of the anime. Are they worth watching or are they dated, horrible Macekres only worth watching for nostalgia (and even that is arguable)? Most fans have a negative view on the dub and mock it however others honestly enjoy it and think it's a good dub (especially for its time period). Then there are those who enjoy them for how hilarious the dubbing, censorship, and voices can be instead of the quality of the dubbed version itself.
Misty's departure pretty much split the anime fandom into those that won't have anything to do with any episodes past Johto and those that openly embrace the newer episodes.
There's also a divide between people who like the anime and people who think it does more harm than good for the games. The latter tend to want an adaptation that's truer to the games, like Pokémon Origins.
Yu-Gi-Oh!: With the fifth series, Yu-Gi-Oh! ARC-V, there was a split between its fans and the fans of Zexal; while the fandom as whole has always discussed which series is "better" to various levels of seriousness, you'll seldom find any consensus reached in between these two fanbases, let alone people who like both series. The reasons vary according to who you ask, but mostly it seems like Arc V fans think of Zexal as the Dork Age of the franchise, while the Zexal fans look down on Arc V for considering it too ambitious or convoluted for their tastes.
The Endless Eight arc of The Melancholy of Haruhi Suzumiya anime has sharply divided the once-strong fanbase. The main problem wasn't that it was adapted, it's that a short "Groundhog Day" Loop story was adapted into 8 full episodes with little variance between them, and that those episodes took up half the season's episode count. You either believe that it was all meaningless Filler that wasted half a season on what was a single, brief story in the novels or a clever Mind Screw that forced the viewer to share Yuki's pain in being the sole person fully aware of each loop for almost 595 years and being unable to do anything about it. Which side you're on also may depend how you first experienced it. Were you a fan that was watching the episodes each week as it was coming out or a fan who came in after the fact and could watch the episodes at their own pace. Unique in that it managed to not only split the fan base, but the entire cast and directing crew as well—the 8 episodes were extremely similar, but different enough (camera angles, characters' outfits, etc.) that they practically had to remake the whole thing seven times. Some people got in serious trouble over Endless Eight and the fan reaction.
Fairy Tail fandom is extremely divided on the subject of Jellal Fernandez. A major player in Erza's backstory, he made his appearance in the Tower of Paradise arc, which concluded with the manga's hundredth chapter, although technically, he'd been around since the second page of the first chapter. He has appeared in every single arc since then, in some form or another, and opinions are very mixed on if this is a good thing or not. One large segment of the fandom thinks that the unfolding drama is interesting, another thinks that he's worn out his welcome and needs to go away for good.
Naruto: Uchiha Itachi's true backstory. At least in the American fanbase, this plot twist split the franchise itself right down the middle to absolutely polarizing extremes. Those who loved the plot twist venerate Itachi as some kind of tragic hero, while those who hate the plot twist continuously paint him in a bad light or make him as Wangst-y and obsessive as possible. The split created some of the strongest sections of the Naruto Hatedom, some of which last to this day.
Slayers: The third anime season, Slayers TRY: in the late 90's/2000's when it came out, it deviated in formula and story from the first two seasons (darker plot, downplayed humor, and one villain to chase as opposed to a multitude), and the main characters become peripheral to Guest-Star Party Member Filia and antagonist Valgaav. Because of this, it's either the most well-loved season or regarded as a piece of crap - it's even referred to as such by some professional reviewers.
Science Ninja Team Gatchaman: Science Ninja Team Gatchaman vs. its multiple English adaptations. While Battle of the Planets is lambasted for turning the franchise into a joke and being one of the early examples of Macekre, there are just as many fans that will defend the flaws present (or love it in spite of such) and insist that it is the superior dubbed adaptation by the fact it came first, had a high budget, and was a source of much '70s childhood nostalgia.