The most prominent example would be the belief that the Joker killed Batman's parents, which stemmed from the 1989 movie. In comic book canon, Batman's parents were either killed by Joe Chill, or an unnamed mugger (depending on the era), not the Joker, who is apparently around the same age as Batman anyway.
In addition, there is, from the same movie, the Joker's name Jack Napier, which is taken as canon for many fans, though he has no confirmed name in the comics (though many homages to that name have been made). "Napier" is actually something of a joke, appropriately enough: "Jack Napier" is intentionally reminiscent of "jackanape", which is to say, a joker; in this it is something of a Prophetic Name. The name is also an homage to Alan Napier, who played Alfred on the '60s TV show and died shortly before the 1989 movie was made.
They toy around with this a lot in the DCAU. In an episode of Batman: The Animated Series one of the psychiatrists at Arkham Asylum describes several Bat-villains by their first names, calling the Joker "Jack Napier". In a couple of other episodes they use it as his name, but in a couple they state it's one of his aliases, so they don't really give any confirmation on whether it's canon in that universe or not.
One comic (of uncertain canonicity) features a cousin of the Joker called "Melvin Reipan". Whether this is actually his name or an alias (or if Joker would even share it, being a cousin) is unknown. He also addresses Joker as "Cousin Ja-" before getting cut off.
The Dark Knight averts this by specifically declaring the Joker has no known real name or any sort of identification.
In "The Origin of the Joker" in Countdown issue 31, one of Joker's possible identities is "a mob killer named Napier".
A particularly odd example is the belief that in The Killing Joke, Barbara Gordon was shot in the vagina. Despite the art clearly showing her being shot in the stomach, and the fact that being shot any lower than that wouldn't have caused her the spinal damage that resulted from the attack.
One result of Babs's disability, is that she could not have children anymore; perhaps that had something to do with it? It probably doesn't help that the whole attack is presented much like a rape.
The Joker Blogs seems to be heading into the realm of fanon, even though it is made by fans.
In the same vein of people's only knowledge of characters coming from movies, there's the idea that Wolverine and Sabretooth of the X-Men mythos are half-brothers, which, while used in the X-Men Origins movie, does not actually apply anywhere else. Though, of course, with Wolverine'spast, who can tell?
There's also the widely-held belief that Dog Logan from the limited series Origin (who was heavily implied to be Wolverine's half-brother, and was used as the basis for the films' portrayal of a young Sabretooth) was actually Sabretooth, even though the author of said comic went on record saying that he never intended this. note Not to mention the rather explicit in-universe evidence to the contrary: Dog's last appearance in the comic has him with three permanent scars on his face from Wolvie's claws (even though Wolvie himself has long since discovered his healing factor by this point) seeming to indicate that he doesn't have Sabretooth's healing powers.
Speaking of Wolverine, there is a popular theory going around that Rachel Summers, a member of the X-Men from a parallel reality, is actually the daughter of Wolverine and X-Man Jean Grey, despite the fact that the comics clearly state that Rachel is the child of her reality's Cyclops and Jean Grey. Despite any evidence to the contrary, this theory is stated as fact by several fans.
There were some hints, at least, that Scott might not be Rachel's father. There's absolutely no evidence, at all, anywhere, that Wolverine is her father. Claremont's original plan, supposedly, was to have Rachel be the daughter of Jean and the Phoenix Force.
A Tabletop Games tie-in for The DCU placed Gotham City in New Jersey and Metropolis in Delaware, but these details are never mentioned in any comic books despite numerous fans accepting it as fact. Metropolis has also been described (if only in our world) as New York in the day and Gotham City as New York at night, but that's for poetic reasons; New York exists as a separate entity.
While Metropolis's location still hasn't been given a canonical placement, there are at least two canon references to Gotham being in New Jersey — Legion of Super-Heroes, pre-Zero Hour, made reference to the Bat-Cave being located (by archaeologists) in the Jersey sector of Metropolis, and an issue of Shadow of the Bat showed a character's driver's license, giving him an address in "Gotham City, NJ".
The "Countdown to Final Crisis" special showed a computer display which listed biographical information about several DC super heroes. "Gotham, NJ" and "Metropolis, DE" were pretty clearly visible on Batman and Superman's files.
Arkham Asylum is only one indicator that Gotham is in Massachusetts, its relationship to Boston analogous to that of Metropolis to Manhattan.
The placement of Arkham Asylum in Massachusetts must be a reference to H.P. Lovecraft. The name "Arkham" comes from a fictional town in Massachusetts where some of his stories are set. That's unfortunate; there must have been some way to make the Shout-Out that didn't require moving Gotham to yet another state...
Interestingly enough, Arkham Asylum's (or Arkham "Hospital's") debut in Batman #258 (1974) explicitly places it in New England... but does nothing to confirm that it's in or even near Gotham. By the time Arkham was confirmed to be in Gotham in Batman #326 (1980), the New England angle was largely forgotten.
Pre-Crisis, Metropolis was sometimes stated to be on the East Coast, sometimes in the Midwest, and sometimes even on the West Coast.
Then you have Smallville. Originally, its location was usually unstated (though one comic put it on the Atlantic Seaboard). It wasn't until the Christopher Reeves film that it was placed in Kansas. Which becomes canon in The Man of Steel since.
The show Smallville threw a wrench in the works by placing Metropolis within sight of Smallville, both of which are stated to be in Kansas. However, Smallville is clearly an adaptation and routinely deviates from other Superman media. Most Superman comics place Metropolis pretty darn close to Smallville, however.
Much like River from Firefly, Gambit often refers to himself in the third person in fanfiction. This happened rarely if at all in the comics. This might, however, have stemmed from the X-Men cartoon of the '90s, in which Gambit did this quite often.
Strange things happen to Gambit in Russian fandom. 1) He is called Creole nearly all the time (he's Cajun in canon). 2) He's always viewed as an extremely Casanova-type character (which he is not in canon or at least not more than other characters in canon; example — Wolverine). 3) He's bisexual. This has been so ingrained in the fandom that most people seriously believe he is bisexual in canon (!) or at least intended to be. 4) Terrence Dash. Just Terrence Dash. Don't ask... this purely fan-made character (Gambit's gay pairing) is treated almost as if he was canon in the first place.
The first two things can be explained by the fact that most of Russian fans of X-Men first got acquainted with them via X-Men, where in dub Gambit indeed was called Creole quite often and constantly tried to charm some casual girls. Wolverine's not getting the same treatment may be connected to the fact that his Girls Of The Week usually played a significant part in the episode's plot and he didn't play up his "ladies' man" image. Two latter seem to be a result of the huge influence of the first generation fans.
People have noted that Squirrel Girl's victories against villainsfar strongerthan her is due to her Plot Armor only working in proportion in how strong her foes are, thus she'd lose if she met regular mooks. However, if you'd ask the Bug-Eyed Voice and examined the nuts of one of the mooks which were defeated by her in GLA Issue #2 you'd notice that she is still overpowered against normal mooks who run screaming from "the bringer of Anti-Life!".
Many recent revivals of Golden Age public domain comic book superheroes wind up incorporating fanon into canon because, well, Golden Age comic books are hard to find, leaving writers no choice but to rely on character profiles found throughout the Internet. Many of those profiles have at least some fanon. In fairness, Golden Age comics were never big on continuity, so lots of this fanon comes from attempts to reconcile contradictory details and justify things that just plain didn't make sense.
Female characters that were savagely beaten and hurt in canon will have been raped according to fanon. Perhaps this is because Rape Is a Special Kind of Evil or people do not understand why a hero being hurt is such a big deal. Some examples of this include Rogue when she was depowered in Genosha, Batgirl (Barbara Gordon) when she was attacked and crippled by the Joker in The Killing Joke, and Black Canary being beaten by a gang in The Longbow Hunters. Word of God has confirmed none of them were raped.
Gwen Stacy is an interesting example because canon wise she was a short tempered Tsundere that Really Gets Around and was continually trying to seduce Peter because he was not interested in her, despite the fact that during that time she was dating Flash Thompson and later Harry Osborn. After she began dating Peter she developed into a truly caring girlfriend. After she was killed by the Green Goblin Fanon disregarded her history and turned her into a Purity Sue Virgin and claimed that it was impossible that she would ever sleep with a man despite the fact she had early on claimed to have slept with every man she knew.
The only thing known for sure about the person who killed (or was otherwise responsible for the death of) the first Despair in The Sandman is that he "will take the rest of eternity to die. Only then will his pain cease." A very popular fan theory is that his fate was to become the second Despair.