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.
Jerk Sue: Logan is supposed to be an asshole, it's a major aspect of his character. But, following Character Development, he became a Jerk with a Heart of Gold and had a tendency towards Jerkass Has a Point, and it's one of the main aspects that made him so popular. But, in recent years when he's being handled by Ascended Fanboys, this tends to be 'showcased' by him ranting about how 'wrong' someone's behaviour is (usually Cyclops), despite the fact he himself has done much worse, is currently doing much worse, or is plotting to do much worse. Take, for example, his arguments in Schism, where basically he compares training the younger students how to fight as training child soldiers, despite the fact that, since their inception, the X-Men have ALWAYS trained young children in how to fight, with Logan in particular making a habit of picking out a young teenaged girl and taking her under his wing. He later goes on to create an entire institute dedicated to teaching kids how to face ninja armies, and learn how to survive being abandoned on the Savage Land or die. Due to this happening far more often in recent comics, Logan has slowly evolved to the point where being a hypocrite has almost became a defining character trait.
Wolverine has his own Complete Monster in The Founder, the Big Bad of Jason Aaron's run. In the 1920s, The Founder was the young son of a notorious robber-baron whose workers went on strike. The father brought in thugs from the Pinkertons to control the strike, killing many workers. The workers hired Wolverine to fight the thugs. In response to this, the robber-baron murdered the head of the union. In responser to that, Wolverine killed the robber-baron, whose son swore revenge and dedicated the rest of his life to learning everything, everything about Wolverine. Over the years he recruited many others with perceived or real grievances against Logan, forming the cult called the Red Right Hand. When a member tried to leave the group, he forced her to smother to death a helpless old woman (Sabretooth's mother, as it turns out). In the modern day the Founder, now an old man, decided to act against Wolverine personally. He and the other members sold their souls to the hell-lord Marduk Kurios, in return for damning Wolverine to Hell and allowing a demon to possess his body. They then let the demon and their bodyguards, the Mongrels, loose on anyone who had ever been kind to Logan, ever (killing John Wraith, Silver Samurai, and temporarily, Mystique). Once Wolverine got out of Hell and back into his body, he chased after the Founder and the Red Right Hand, killing the Mongrels along the way, only to find that the cult had already committed suicide. The Founder left a tape-recorded confession for Wolverine, though, in which he revealed that he never intended Wolverine to die, oh no. He wanted him to win. Why? Because the Mongrels were Wolverine’s long-forgotten illegitimate children. Even though Wolverine defeated his enemies, the bad guy decidedly won. In the narration for his flashback (provided by him), he mentioned how much he enjoyed what an evil man his father was. To him, the ability to brutalize others is a sign that a person is "strong."
Tomi Shishido, the Gorgon, fully lives up to his name by turning people (especially girls) into living statues, which he uses for decoration. His villainous career started very early, when he founded the Dawn of the White Light, a mutant surpremacy group which put even the Brotherhood to shame. At the age of 18, he slaughtered his entire family in order to prove himself worthy of joining the Hand. His by far most heinous act was when he brainwashed Wolverine into a servant of HYDRA and sent him on a brutal killing spree, which caused several dozen innocent deaths, among them the Hornet and Northstar, which haunts Logan to the present day. He also has absolutely no problems with killing children, which he showed when murdering Phobos.
From the Wolverine: The Best There Is limited series comes the enigmatic Mr. Winsor, alias Contagion. Born from a Latverian eugenics program, Winsor's body contained every single disease known to man, which he trained himself to be able to release at will. Killing all the other children born from the program, Winsor struck out on his own, acquiring the services of numerous nearly-immortal individuals as both enforcers and test subjects, whom he would infect over and over again with his diseases in order to make them stronger and more deadly. Winsor used his abused son, deformed from exposure to his father's diseases, as bait to acquire Wolverine for his experiments under pretense of wanting to use his research to cure his son's condition, knowing Logan, as a Friend to All Children, would never refuse the offer. Ultimately, however, he revealed his true motive: in making his diseases stronger and stronger he hoped to engineer one that could kill absolutely anything, which he would then unleash on the world and then the entire universe via a stolen spaceship. When his followers ceased being useful, he subjected then to And I Must Scream to ensure they would not rebel. With evil plans unmatched in scale for an Earth-based villain, Contagion more than makes up for his brief tenure in the comics with his cruelty.
It's a trend which no adaptation of the X-Men has escaped. Any issue with the lineup of them, and Wolverine will be front and center. There was even a cartoon series called Wolverine and the X-Men, where the X-Men got second billing. The first three X-Men films (especially the third one) are arguably more like Wolverine films than X-Men films.
Worth noting one series, X-Men: Evolution, escaped this trend. While Logan was an important character in it, the show focused mostly on the teens.
Another card-related one was the tie-in Top Trumps-like card game for the third film (one which arguably solidifies Logan as the Creator's Pet). Each card had four traits, Speed, Strength, Fighting Ability, and Intelligence, each ranking one to seven. Most characters had about two for two of the traits, a four for one more, and six or seven (e.g., Angel had lower traits for the latter three, but a six in speed, while Mystique had low for all but fighting skills and intelligence, and Colossus excelled at strength). Logan had six or seven for every trait. He's apparently a better fighter than Mystique (which as the first X-Men movie showed was not the case), faster than Angel, stronger than Colossus, and smarter than Doctor Doom.
Part of what went wrong with Logan is simply that he has been so overused that he has no character core anymore. So many writers have used him, and played mainly on his appeal to the fantasies of young teen males who think the caricature version is kewl, and his healing factor and other powers so used and misused, that there's nothing much left.
After the events of Schism, he got his own team of X-Men separate from Cyclops' group (though he did lose his solo title around then as well). He's been set up as the successor to Professor X, establishing a new school and everything.
He tends to be pretty broken in video games too. In Marvel PuzzleQuest, his (and Thor's, oddly enough) 2 star comics are absolutely devastating, being able to place a strike tile (added damage for every attack from all team members) for every 5 red you have at the time and able to do 70% more damage with his red skill than Iron Man's for only 2 more red. Did we mention that strike tiles work with abilities too? And he's difficult to take out quickly, just because there are X yellow tiles on the board keeping him alive. Even Daken, who's also known for being a tough enemy pales in comparison, despite not actually having to activate his 100% passive abilities. And then there's the 3 and 4 star comics.
Dork Age: Arguably the "bone claws" arc, but the "feral subhuman" arc (after Genesis' failed attempt to restore his adamantium) is probably closer to the trope.
Jeph Loeb's run. When it was announced that there would be an animated movie based on his run, the reaction was a) Nobody wants this and b) The animation sucks. It doesn't help that Lodb introduced a Villain Sue in his run.
Internet Backdraft: Whereas the Bishounen-style Tony Stark in the upcoming Anime-adaptation of Iron Man is met with almost unanimous praise from even long-time fans, when an attempt to apply the same style to Wolverine, the sheer tidal-wave of revolting Asian-hate from Marvel fans that flooded the internet was amazing and sickening to behold.
To be fair, Wolverine is known for being short and rugged looking, a direct contrast to a Bishounen, still the reaction was over the top.
Moral Event Horizon: Sabretooth crossed this long, long ago, but (for Logan) it was when he beat, mauled, raped and killed Silver Fox, then left her body in her and Logan's cabin for Logan to find. What really puts Sabretooth over the top is that Logan immediately ran off for a Roaring Rampage of Revenge. Which, once he found him, he didn't win... The incident turned out to be a faked memory that was implanted in Logan, Sabretooth, and Silver Fox by the Weapon X program... and then Sabretooth ended up killing Silver Fox anyway.
The big one for Sabretooth was when Tieri wrote him raiding a daycare center and casually eating the terrified children as snack food.
Creed now has yet another truly heinous act under his belt in the form of feigning affection towards Daken and playing the role of the father he never had so he could slowly groom him for a fatal confrontation with Wolverine which did indeed come to fruition, forcing him to kill his own son. Also, he obviously never actually meant any of the kindness he showed to Daken and would have killed him himself if the plan didn't take. Why did Creed do this? Simple. He just wanted to hurt Logan on a level far deeper than he could ever manage on his own, and he succeeded.
My Real Daddy: Len Wein may have created him originally, but Claremont (and partially John Byrne also,) made Wolverine the man we know today. Even after Byrne left though, Claremont kept adding to the character, specifically a deep love and appreciation of Japanese culture, as well as many other things. (Such as Madripoor.)
The Scrappy: Romulus. A massive Villain Sue who Jeph Loeb keeps using, even though fans hate him. Hell, on CBR's review of the final confrontation between the two, it actually says "and let's never speak of him again". The character is not liked.
Squick: Some of what happens to Wolverine, as well as stuff he's done to himself, is just stomach-turning.
Getting the adamantium yanked out of him was one of the most gruesome moments of the 90s.
Then again, given she specifically called him "James", the likeliest scenarios would either involve time travel, or them hooking up soon after he learned his real name, in which case she'd still be 18.
Any time he uses his claws to cut up and serve food to other people, given that his claws are kept inside his arm.
Not seen, but mentioned by him: while trapped under a glacier for six months, he survived by slicing off and eating pieces of his arm.
Suetiful All Along: Originally (and sometimes nowadays as well) he was constantly described and depicted as physically ugly, having a distinct aversion to bathing, and shorter than most teenage girls. He's also got a personality so abrasive it's a miracle he ever lasted as part of a team in the first place. Unfortunately, he speaks 17 languages, his powers make him functionally immortal (he's at least 114 years old), he has two adoptive daughters, he's been married twice, and he's had more romantic relationships than any other X-Man, in spite of the negative qualities listed above. In adaptations, he exemplifies the Black Hole factor of sues - other X-Men are Demoted to Extra so he can take all their roles (and love interest, in Scott's case.)
Villain Sue: Not Logan himself, but in his solo series, Jeph Loeb introduced a character called Romulus... whose Villain Sue-ness has reached memetic levels. He is retconned into being influential in many parts of Wolverine's life, and in fact, has been manipulating Logan for his entire life. He apparently has enough connections to make the Russians use the Winter friggin' Soldier to assassinate Daken's mother, and has a really dumb backstory that somehow links all mutants who have canine-esque mutations, and also says that he is several thousand years old. Oh, and he has subtle telepathy which he can use on Wolverine to create false memories and bring up repressed ones! It also seems like Loeb tried to have Romulus replace Sabretooth as Logan's Arch-Enemy.