Wolverine. People either think he's the most badass and complex character in the X-Men lineup or an overhyped, overpowered Jerkass who gets way too much spotlight at the expense of other characters (especially Cyclops).
Apocalypse is another big one, and it's one that seems very dependent on when you started following the comics. Fans who started following the comics around the time he was introduced likely think he's an incredibly badass villain, one of the all-time greats, and easily the biggest threat to the X-Men, with a neat backstory and some pivotal stories to boot. Fans who started following significantly before or after his introduction tend to consider him a laughable villain who, despite his ridiculously unlimited powers and massive advantages, hasn't actually accomplished anything besides giving the heroes more powers, with a look ripped off from Darkseid and Thanos to boot.
While not strictly speaking an X-Men villain, Thanos is this to a segment of the X-Men fandom for the exact same reasons described above for Apocalypse: laughably overpowered, Story-Breaker Power with nothing to show for it, a lamer proto-version of their villain of choice and... yeah. The rivalry between Apocalypse and Thanos fans, while simmering in the background compared to the front-and-center Cyclops VS Wolverine argument, is most definitely this.
Kwannon, or rather Kwannon as the new Psylocke, has proven singularly divisive. Some fans see her as nothing more than a Replacement Goldfish, as it has been outright admitted that she has been brought back and made into the new Psylocke so fans can have a version of the "sexy Asian ninja" Psylocke they have known and loved for the past 30 years, even if she's a completely different character mentally. Other fans have embraced this change wholeheartedly, arguing that with the Unfortunate Implications of Kwannon and Psylocke's original story and the time for Kwannon to be restored to the body that was originally hers is long past overdue and that, being a blank slate in terms of characters, writers have plenty of opportunities to establish her differences from Betsy. As this change is so new at the time of this writing (2020), only time will tell how well she is received in the long run.
Wanda, aka Scarlet Witch, is an odd case as her base breaker status comes down to whether one is more a fan of The Avengers or the X-Men. Avengers fans have largely forgiven her for her actions during the Disassembled and House of M arcs, mainly because it was established in Avengers: The Children's Crusade that she was not herself at that time and even with X-Men fans, she's quite popular with female comic book readers and those with mental illnesses. On the other hand, for most parts of the X-Men fandom, she is viewed as a Karma Houdini who never got any real punishment for her actions in those aforementioned stories. One particular sore spot is that while Cyclops (whose actions also happened under a mental breakdown and possession) gets hunted down like a criminal, Wanda has largely been forgiven. This has been pointed out In-Universe on occasion. Her attempts to fix her mistakes have often been aborted or something wrong happens during the attempts, with fans either saying she should be punished even more or are annoyed that the writers are pulling an invoked Ron the Death Eater for her good intentions.
Beast. Starting with Second Coming, his increasing jackassery has moved him into this territory. Some are still nostalgic for his more charming moments while others have grown beyond tired of his hypocrisy and want him to go away.
Colossus, albeit he's a bit more of a downplayed example compared to Cyclops, Wolverine, Emma Frost, and Jean Grey. While Colossus has his fans because of his Gentle Giant personality and perseverance in the face of tragedy, there are fans who also dislike him for a number of reasons, mostly revolving around his relationship with Kitty. Detractors dislike the original age gap between him Kitty, him leaving Kitty for Zsajia, beating up Pete Ross when he found out he and Kitty were dating, and feel that his and Kitty's relationship often infantilizes her and has been Strangled by the Red String. Reasons for disliking Piotr not connected to his relationship with Kitty are his joining Magneto's Acolytes after Illyana's death and his naivety regarding Illyana's changes after her experiences.
Vulcan. For some fans, he is the worst part of whatever story he's in, which adds unneeded darkness to Xavier's character. They also find him too whiny and obnoxious to be taken seriously, while being powerful in a boring way. To others, he made up for this in his depiction in Emperor Vulcan, The Rise and Fall of the Shi'ar Empire and War of Kings.
Emma Frost always has been a divisive character ever since she reformed and joined the X-Men. She started out as a villain but has not exactly changed her attitude. Her romantic relationship with Cyclops is despised by those that prefer him with Jean Grey. While she has a considerable number of detractors, her fanbase is just as large and vocal. This trope took on a whole new level after her Face–Heel Turn in Inhumans vs. X-Men where she turns into a genocidal madwoman bent on getting revenge for her beloved's death caused by the Terrigen Mists against the Inhumans. Fans are torn on whether her actions were justified or it was just plain bad writing that undid two decades of Character Development, or if her betrayal was the only logical path for a long-time villainess that never truly reformed according to her detractors. However, she's been considerably less divisive since House of X, being mostly well (and consistently) written.
Gambit is another very polarizing character.
For some, he is Cool Personified, what with his accent, Badass Longcoat, playing card motif, mysterious origins, tragic backstory, and suave demeanor, and one of the most stylish and interesting characters in the entire franchise. His fans also love him for refusing to give up on his relationship with Rogue and find his dedication to her touching and sweet.
For others, he is a living personification of everything that was wrong with the The '90s. His backstory is convoluted, his costume is ridiculous, his Funetik Aksent makes it hard to read his lines (or just plain sounds stupid), his entire characterization is so overly focused on mystery and "style" that he comes across as flat, and his powers are useless and stupid. He is also blamed for "ruining" Rogue by giving her a reason to angst, holding her back from any real development, and just milking the soap opera cliches for all they are worth.
A third group hates him for an entirely different reason, as his secondary power (superhuman charm - which, admittedly, is rarely discussed these days) and the way he uses it makes him a borderline sex criminal. His habit of ambush-kissing various X-ladies (regardless of who is or isn't in a relationship at the time) does not help, nor does the time he very nearly slept with a teenager (well, Mystique disguised as a teenager, but he did not know that).
Lately a 4th group popped up who loves him for the sheer narm of the character, how hard they try to play him as cool despite some of his less than pleasant traits, his purple costume, his exaggerated accent, and just how much the 90s is a part of him. To many, Gambit is so dumb it is impossible to hate him.
Jubilee is possibly one of the biggest out of all of the X-Men characters when it comes to fan opinions. She is either widely loathed and considered The Scrappy (specifically a Replacement Scrappy for the much-beloved Kitty Pryde) or very much beloved and viewed as an Ensemble Dark Horse. Both sides tend to be equally passionate in their views.
Her powers are this in of themselves, with many dismissing her as useless while others point out the books have shown that she is much more powerful than her fireworks would suggest.
Kitty Pryde. It pretty much depends on when you started reading. For many who grew up with her, she is a badass who grew into an X-Man in her own right and is the ideal of what a young mutant character should be. For others, she is a bland character who only gets focus because of those Running the Asylum, and there are much more interesting characters who deserve more focus.
Cyclops. It's not helped by the fact that the writers themselves are split on if they like him or not.
Especially when he and his group of X-Men became wanted fugitives who are hunted by the Avengers, it led to a large split between fans who think he's making things worse by opposing SHIELD or if his intentions are well enough to justify his actions. It's telling, that whenever Cyclops gets into an argument with someone now, half the fanbase responds by complaining about what a dick the other person is being, while the other half complains about him instead.
The announcement that he's essentially responsible for about 50% of the bad situation that was affecting mutant-kind (the other 50% is Terrigen Mists being now lethal to mutants, but they didn't pissed off humanity back to "Kill All Mutants" mode) had many people wanting the X-Men to find him so they can lynch him, and others wished to at least wait until he's found so he can give his defense... And then Death of X came out where it was revealed there was not one but two Terrigen clouds, one of which hit Muir Island and killed Jamie Madrox as well as all of his dupes, this being the first time any of the X-Men had seen M-Pox and being understandably horrified. Cyclops begins a plan to use a mutant named Alchemy to change the make up of one of the clouds and render it harmless to mutants which works although this leads to Alchemy's death, all with the Inhumans fighting him every step of the way. He's then executed by Black Bolt in front of the gathered X-Men and Inhumans. That leaves the reader to wonder just why, then, was he being talked about as having done something so irredeemably evil. Then Inhumans vs. X-Men revealed that he died sometime around the first issue of Death of X and all instances of Cyclops afterwards are Emma Frost's mental projections, meaning he didn't even do the crimes he was being accused of.
Teen Jean Grey in All-New X-Men. While most people find her interesting to read, how much they actually like her varies, due to her very abusive use of her powers. Thankfully, she gets better.
Regular adult Jean Grey herself, as well as the versions of her from popular adaptations, also tend to get this. Jean is either seen an uncompromisingly compassionate woman with a fiery temper and occasionally has an uncontrollable Superpowered Evil Side, but ultimately acts as The Heart of the X-Men, 'OR she's a boring Invincible Hero goody-two-shoes with no flaws or personality beyond 'she's nice'. In X-Men: Evolution, the 'nice and flawless' interpretation is replaced with being an Alpha Bitch in the eyes of some fans (especially those who preferred Rogue), though that is more explicitly a case of Ron the Death Eater; she was more Spoiled Sweet.
Sabretooth. The direct reason for his Badass Decay is listed above. Originally a forgettable-if-badass villain (so much so that plans to make Sabretooth and Wolverine related were nixed because editors saw Creed as a low-rent henchman not on Logan's popularity level), Sabretooth's popularity exploded during the 90's. This was a period when any character could get their own book if they were popular enough, and due to his Badass nature Sabes did get a few limited issue runs, playing him as a Villain Protagonist while still giving him some much-needed Character Development. This didn't sit well with some fans, who were offended by Creed's Politically Incorrect Villain moments and the Comedic Sociopathy that came from runs written around him (or even just storylines in which he appeared). Starting with Chuck Austen (the writer of the "beaten up Sabretooth with a black eye being laughed at" story above), these fans started Running the Asylum and trying to neuter Creed's popularity by writing him as incompetently and as pathetic a figure as possible. He was even Killed Off for Real at one point, only for Joker Immunity to kick in as he was far too popular and prominent a character by that time to be killed off permanently. As of 2018 the base is broken, perhaps irrevocably, between those fans who look back on the 90's fondly and consider Sabretooth as a cornerstone character of both the X-Men and Wolverine rogues galleries, and those fans who hate the 90's and regard Sabretooth purely as a Hate Sink who should never appear in a storyline again (or if he must appear, at least be written as much of a Butt-Monkey as possible).
outright admitted
that she has been brought back and made into the new Psylocke so fans can have a version of the "sexy Asian ninja" Psylocke they have known and loved for the past 30 years, even if she's a completely different character mentally. Other fans have embraced this change wholeheartedly, arguing that with the Unfortunate Implications of Kwannon and Psylocke's original story and the time for Kwannon to be restored to the body that was originally hers is long past overdue and that, being a blank slate in terms of characters, writers have plenty of opportunities to establish her differences from Betsy. As this change is so new at the time of this writing (2020), only time will tell how well she is received in the long run.