This was thoroughly mocked in the Potter Puppet Pals episode "Draco Puppet", where Harry responds to the numerous fan requests to introduce a Puppet Pal Draco. Having established that he's doing this, Harry spends the rest of the video in Sarcasm Mode as he introduces us to our new puppet friend:
Another slightly subtle example on Draco Malfoy's at times almost saintly status in fanon is this: In the canon, i.e. in the books, Harry always calls him and thinks of him as "Malfoy". Harry almost always thinks of teachers and authority figures (which Malfoy can't exactly be categorized as) using their last names, and the same goes for other people he disdain and/or feel aloof to.note Rubeus Hagrid is the sole exception to this, but everyone refers to him by his last name, and he becomes a Professor in Prisoner of Azkaban. However, fans - especially the ones that like Malfoy - usually refer to him as "Draco", while they don't refer to, say, Dumbledore as "Albus" quite as often. Needless to say, there's a reason this trope isn't called "Malfoy in Leather Pants".
It amuses me. It honestly amuses me. People have been waxing lyrical [in letters] about Draco Malfoy, and I think that's the only time when it stopped amusing me and started almost worrying me. I'm trying to clearly distinguish between Tom Felton, who is a good looking young boy, and Draco, who, whatever he looks like, is not a nice man. It’s a romantic, but unhealthy, and unfortunately all too common delusion of — delusion, there you go — of girls, and you [nods to Melissa] will know this, that they are going to change someone. And that persists through many women's lives, till their death bed, and it is uncomfortable and unhealthy and it actually worried me a little bit, to see young girls swearing undying devotion to this really imperfect character, because there must be an element in there, that "I'd be the one who [changes him]." I mean, I understand the psychology of it, but it is pretty unhealthy. So, a couple of times I have written back, possibly quite sharply, saying [Laughter], "You want to rethink your priorities here."
J.K. Rowling has spoken out against people who dare to suggest that Voldemort isn't as horrible as she insists he is. Significant parts of Book 6 are meant as Take Thats against those people. For example, he lived in an orphanage, all right (just like Little Orphan Annie), but the most horrible thing in said orphanage was... himself. He used his magic to bully other kids and was feared both by them and by the staff. It's clear that he was never a good guy.
My Immortal at one point does, in fact, feature Draco in leather pants. So many fanfics portray Draco as a sensitive guy, the opposite of his canon personality, that it's a cliché in itself, but My Immortal goes so far as to make him so ridiculously needy that it seems he would fall straight into Wangst without encouraging words from Enoby. The most (in)famous fanfic example is probably The Draco Trilogy. See that work's page for more.
My Immortal also does the same to Trelawey and Sinistra, in the gestalt "character" known as Professor Sinatra/Trevolry (yes, the names are interchangeable). While neither woman was outright "evil", it's made very clear Sinistra was at best unpleasant and Trelawey cowardly or even downright pathetic, so seeing someone inspired by them being pretty much the "coolest fucking teacher" is jarring.
Averted, even inverted, with Draco's dad, Lucius, whose fans tend to appreciate him for the aristocratic, pimp cane accessorized bastard that he is without giving him the "Awww, he's really just a sexy Woobie!" treatment. In fact, Lucius provides an easy way for fanfic writers to portray Draco sympathetically. After all, if Lucius is an Abusive Parent, then Draco has a nice Freudian Excuse. And your story has a built-in villain, too! Ironically, in the canon, the fact that Lucius cares about his wife and son seems to be his only redeeming quality. Also, Lucius had no capacity to influence Draco's becoming a Death Eater one way or another, seeing as he was in Azkaban for the sixth book, which was when that happened. Lucius is hardly Dad of the Year, but neither is he the abusive monster who exists to torture Draco.
The "feminist fangirls" who refer to Bellatrix as a "role model" because she almost kills a young girl and then mocks her very pissed off Mama Bear upon both that and the death of one of her sons. And then they bash said mother for killing their Bella. Never mind that Bellatrix is a Yandere who only has eyes for a man who doesn't give a damn for her, or that she's openly said that if she had kids she'd sacrifice them to Voldemort, or that she was among the Death Eaters that tortured Neville Longbottom's parents into insanity — she's a "super great feminist" for bashing motherhood. Unfortunate Implications, much?
Finally, if a Peggy Sue is involved (either Bellatrix herself or another character), she may be steered away from the person she becomes in canon. Where it goes from there varies as much as the fanfictions written about her—Anti-Hero, Anti-Villain, Mary Sue, Author Avatar...
Inadvertently done in the film adaptations with Narcissa Malfoy, mainly due to the Quidditch Cup (which established her as well-bred yet unpleasant and elitist) being cut. Her first appearance, in Half-Blood Prince, is now entirely sympathetic—that of a terrified mother attempting to keep her only son from getting killed.
Salazar Slytherin had a dream that one day, an adequately racist and motivated student would find the secret underground chamber he built to house a deadly basilisk and use it to mass-murder his Muggle-born classmates. And some fans would have you believe that's just dandy. (This one has actually been done well in some cases — Slytherin lived at least a thousand years before the series takes place and conceivably could have been the victim of an in-universe Historical Villain Upgrade.) It helps that the series implies that he wasn't always a "twisted old loony". In fact, according to the Sorting Hat, he and Godric Gryffindor had once been the best of friends.
A good example of making him sympathetic without contradicting canon can be found here. Adult wizards had all kinds of magic they could cast to survive being hunted and burned, but families were prone to losing younger members. Girls could be burned as witches at nine and boys at ten, well before they were able to defend themselves. Children of Wizarding families had some protection in their parents and older siblings, but Muggleborns were SOL and often turned over by their own families. A thousand years have twisted his reputation into that of a bigot and a psychopath instead of the intelligent and cautious man he was (and is, as a ghost).
Severus Snape gets put in leather pants a lot by fangirls. There are the ones who want to know him carnally, and then there's the fans who have blamed Lily for Snape calling her "Mudblood", or even tried to justify it and deny its connections to real-world racism. Oh, and they also bash Lily for "not being a good friend" (aka, refusing to become Snape's Extreme DoormatLove Martyr) to the guy who called her racial epithets. Snapefen have even gone as far as to make snide comments about JKR's abusive first marriage, such as comparing her to a brood mare. Anything we could say about the Snape-dom can be summed up here.
It doesn't help any that Snape's misdeeds (as a child, anyway) are continually downplayed in the novel and he's portrayed as a kid with a really abusive, racist family - in other words, a first-class Woobie. The situation gets even more extreme when you consider that he's regularly contrasted with the any-house-other-than-Gryffindor-hating, shallow, vain, bullying jockish brat that is the young James Potter. Rowling makes it really overt that James, Snape's romantic rival, behaves just like Malfoy, even saying with regard to being in Slytherin "I think I'd leave, wouldn't you?", which is exactly what Draco says to Harry about Hufflepuff in the first book. It's so deliberate on Rowling's part...
Scabior the Snatcher in Harry Potter and the Deathly Hallows has become this because of his portrayal by Nick Moran in the film. The number of women wishing they could trade places with Hermione and have their hair sniffed by him is growing.
Also, Petunia Dursley. Yes, the JerkassResentful Guardian and Stepford Smiler that abused Harry ever since he went to live with her family, is sometimes shown as a poor victim "bullied" and "insulted" by her "stupid and snobbish" sister Lily, thus she "can't help" but hate Harry and blame him for "Lily's sins".
In the case of the Dursleys, both Rowling and the fandom agrees that Vernon is the least sympathetic of the three. Rowling never gives him a Freudian Excuse and fanfics often portray him as pure evil. However, while the fans seem to consider Petunia the most sympathetic Dursley, Rowling seems to consider that to be Dudley, feeling that his actions were the result of bad parenting. Prior to the end of the series, fans guessed that Petunia would eventually apologize to Harry, but, to the surprise of many, it was Dudley who apologized while Petunia only got an implied Ignored Epiphany. Dumbledore himself considered Dudley pitiful and called out his parents for raising him the way they did - as a selfish, entitled bully that thought he could get away with anything until he was a teenager.
Barty Crouch, Jr. due to being played by David Tennant in the movie and having a Tragic VillainBack Story in the book. Sometimes he gets paired with characters nearly twenty years younger than him like Hermione or Luna.
Interestingly played with in the fic Methods Of Rationality—Draco is a genuinely likable boy who just happened to be raised by Darth Vader. Harry interacts with him rather differently in his first meeting and instead of the immediate politically charged rivalry, we get a less immediate yet still politically charged kind of alliance. And then later on, when he's an occasional viewpoint character with a fairly sympathetic perspective, you keep running into the fact that he believes Dumbledore to be entirely evil.
Fenrir Greyback, a cannibalistic werewolf in canon, is often portrayed as a big protective dog who's only looking out for the best interests of his clan/mate (sometimes Hermione).
Marrietta Edgecombe gets a bit of this as well. Admittedly, we don't know that much about her, but some fans like to imagine her as a tragic heroine who is torn between loyalty to her family and the DA, and that her punishment is completely unfair. Except she's shown to not want to be in the DA from the start, only there because Cho practically dragged her to the first meeting. People also like to play up that she's scarred for life when she never even seems to consider going to Hermione and apologizing for what she did in the hopes that she'd remove the scars. Considering that the school healer couldn't clear them off, why would it be reasonable to expect the person who gave them to her to be willing to? Hermione might just alter them into something worse. Also, that assumes that she is sorry, and if she was sorry, was willing to unbend enough to apologize.
The Death Eaters, despite being Magic Nazis, are sometimes considered Well Intentioned Extremists who were simply trying to unify the wizarding world to fight off the inevitable invasion when Muggles discover them. Except there's nothing in the books to suggest that a war would be inevitable even with discovery. In fact, it all but says the war with Voldemort would have gone a lot better if they had just gotten Muggles to help. Plus the DE doesn't actually seem to care about any hypothetical war with the Muggles, except maybe for the one they plan to start and how they want to win it.
Draco gets a fairly interesting variation on this in The Prayer Warriors. Draco is a "secret Christian" at Hogwarts, gets baptized and joins the heroes, and objects to the plan to have Ebony sleep with Harry to get his key, but joins the Prayer Warriors' crusade to exterminate nonbelievers, wants to kill Dumbledore merely for allowing gay sex, and in a later installment, threatens to kill people who don't go along with one of his plans to kill Wawa, the Titan of Water. In short, he's out of character, but interestingly enough, proves to be an even worse person than in canon despite (or perhaps because) of the author's attempts to make him seem more heroic.