Harry Kim from Star Trek: Voyager and his habit of dying and coming back to life. It's been exaggerated by the fans, though, to the point where someone who only knew the show through its fans would think Harry's grave says "Harry Kim: Born: 2341. Died: 2371, 2372, 2373, 2374..."
An in-universe example is his habit of constantly falling for women he can't get. It gets to the point where every time he starts a relationship, his buddy Tom Paris goes off on a litany of every doomed romance he's started in his time on the ship.
Stargate Atlantis: Rodney McKay blew up a solar system (though he'd like to remind you that it was actually just five sixths of it). Stargate Universe: Nicholas Rush dialed an untested address into a gate, marooning him and most of his coworkers on an ancient ship. Let's just assume that being hired as a scientist by Stargate Command requires high knowledge of Stuff Blowing Up.
A recurring joke about the Daleks in Doctor Who was their inability to go up stairs. This weakness was addressed in the 1988 serial 'Remembrance of the Daleks', but the jokes persisted at least up until 2005, when 'Dalek' — which also addressed this point — was shown. It's perhaps worth pointing out that, by the time the first story aired, the audience of Doctor Who was roughly three guys and a dog, so it's possible that not enough people actually saw it for the change to sink in.
It was addressed before then - one earlier episode showed them to have somehow got up a staircase without it actually being shown on screen.
And in some quarters it's still what the Daleks are most famous for, despite the fact that the Daleks, in at least the Russell T Davies era, spend half their time flying around like nobody's business. (A large portion of the Daleks' appearances in the Steven Moffat era have been inside the Dalek Asylum and on board a Dalek spaceship, though hovering Daleks still make appearances.)
When Donna Noble first appeared in 'The Runaway Bride', she slapped the Doctor twice (once because she thought she'd been drugged and kidnapped, another time because he was being rather deriding of that). Since then, she's never slapped him again, and went through a lot of Character Development to become the Doctor's moral compass. Her main characteristic in fanfiction is slapping everyone. Well, that and her memory loss. That particular plot point created the 'Donna Fix-It' fic.
The Sixth Doctor trying to choke his companion Peri to death after a dodgy regeneration. It's still what many fans remember him for.
The Tenth Doctor's last words; "I don't want to go." A Tear Jerker for some but Narm for others.
It didn't help that this was preceded by Ten being completely horrified of his impending death, acting like he would really die and the next Doctor would be someone entirely different, something none of the previous Doctors did.
Similarly, any discussion of "Classic" (1963-89) Doctor Who will feature a lot of people talking about "shaky sets and monsters made of bubblewrap". One particularly dedicated fan has watched every episode of Classic Who available on DVD (which includes most of the Jon Pertwee/Tom Baker era that most people are recalling), and counted exactly one incident of each of those.
The Doctor, especially the Third, is often referenced by his supposed catchphrase: "Reverse the polarity of the neutron flow!" Which the Third Doctor said exactly two times, eleven years apart, as well as once more in a play. The Fourth and a clone of the Eleventh (going through other phrases of his incarnations) used it once and the Fifth and the Tenth twice.
The many fans who consider Adric The Scrappy will never, ever, stop talking about the moment in "Four To Doomsday" where the villain Monarch convinces him to support technocratic dictatorship in about three minutes of conversation. This has been blown up into Flanderisation of him "always siding with the villain", even though the only other times it might be claimed to have happened were two obvious attempts to become a Reverse Mole and one when it was very clearly against his will.
Rory has been referred to in-show as "the man who dies and dies again." He's only done it for real once. He's just... very good at creating the illusion of death. By accident. Lots.
Rose Tyler eats lots of chips. She doesn't, they're mentioned in her first two episodes, then she eats some in her first finale and School Reunion. But fans are convinced.
The Swedish TV show Hipp! Hipp! featured the character Mike Higgins. Despite using the same entry-line in every episode he appeared, it's the last line he ever said in the series that people remember:
Mike: And let me just finish by saying: Go to hell.
One of the biggest pricing game flops on The Price Is Right was a mid-90s game called "Split Decision". It has a reputation for being the game where nothing worked right and the board was constantly falling apart. In truth, there was one playing where two of the numbers fell off their markers. The game's short life was due to the fact that contestants simply had trouble understanding the rules.
In The Brady Bunch, Jan only said "Marcia, Marcia, Marcia!" ONCE in the whole series.
Similarly, in Friends, Chandler utters his famous catchphrase "Could x be more y?" maybe twice. Yet somehow, it becomes the go-to tease whenever any of the other characters mock him.
Lampshaded in one of the last seasons when Joey goes to Monica's Halloween party as Chandler, and his "impression" consists of nothing more than repeating the last thing Chandler said and adding, "BLEARGHHHHHHHHH!" on the end. Everyone laughs, except Chandler (who points out that he doesn't do that).
Ross was afraid that getting divorced a third time would become his Never Live It Down, so for several episodes he didn't tell Rachel that he hadn't filed for an annulment.
Ross getting divorced eventually did become the one thing anyone commented on whenever he was interested in someone. Granted, getting divorced 3 times in the span of 5 years is no easy feat.
Star Trek's James Kirk is well known as a space-traveling playboy who has more notches than bedpost. In fact, despite liberal use of soft lens and hey-look-a-pretty-girl music, Kirk didn't get involved with many of the women he met in his travels, and when he did, it was usually because a) he needed something, or b) alien sex pollen/some sort of mental control. The sole exception to this is Edith Keeler, whom he genuinely did have feelings for. He was even occasionally portrayed as an uptight boy scout, and he never slept with a member of his crew. And the rest of the male cast seemed to get involved with women at least as often as Kirk did.
Although let's not forget "Wink of an Eye" in which after the commercial break the leader of the aliens is combing her hair and the good captain is on the bed putting on his boots.
He does seem to fall pretty hard for android woman Rayna, to the point that when he accidentally causes her death he's sunk into such a pit of depression that Spock ends up Mind Raping him into forgetting her. ('Cos, hey, Status Quo Is God.)
Oh, and let's not forget that half the time that seductive smile was directed at Spock.
Similarly, we don't actually see Scotty drunk or drinking that often, and the most stand-out example was him trying to put one over on an enemy, but in fanfic he's the guy you go to for booze. (The fact that the character is Scottish probably adds to this.)
The funniest part about this is that more often than not Dr McCoy is the one with all the booze.
There's also his boisterous appreciation of the bellydancing in "Wolf in the Fold" and his being the only Original Series crewmember to start a barfight ("Trouble with Tribbles"). That probably adds to his interpretation as the party animal of the original Enterprise.
In a somewhat odd example, Chekov's made-in-Russia bit never became the full-blown running gag it was originally meant to be, but the way the fans go on you wouldn't know better without watching the series yourself.
An in-wiki example - throw a metaphorical dart at the examples featuring Deep Space Nine's Kira Nerys, and there's a fairly good chance the example you hit will mention the fact that she once beat the shit out of a serial killer while the equivalent of nine months pregnant.
The franchise as a whole has yet to live down casting a black guy as the black ranger and an Asian gal as the yellow ranger in the first season. This was actually the result of a last-minute casting change, and it took everyone several episodes to realize the Unfortunate Implications. But that didn't stop them from changing Native American Tommy into the red ranger for Zeo.
On Leverage, Hardison has this over the time he was kidnapped by the Russian mob while pretending to be Parker(the world's greatest thief). Parker herself had the incident in which she stabbed a mark with a fork while she was supposed to be getting information from him.
Megumi Misaki/Blue Dolphin of Choujuu Sentai Liveman is brought to tears in the first two episodes as nearly everything she's known and loved is destroyed all around her by three former friends turned evil. She hardly ever cries after that, yet fandom seems to believe she did so nearly every episode thereafter.
Also, Dai Sentai Goggle Five is actually a show based on science, about good science vs bad science in order to create a better future. But because their weapons are based on gymnastic apparatus (and only one of them (the sole girl of the team, Miki Momozono/Goggle Pink) is the actual gymnast), they were mistakenly thought as a team of rhythmic gymnasts and every single Sentai references are going will remember them as a gymnastic team (their actual jobs, aside of Miki, are as follows: Explorer, chess player, hockey athlete and zoo worker). Even Kaizoku Sentai Gokaiger makes fun of it!
In-universe example in Law & Order: Special Victims Unit: A social worker is called to trial for placing a child in an abusive home, which eventually led to his death. The bad publicity and death threats she receives drive her to suicide, with her last words lampshading that she has saved hundreds of children but will only be remembered for this one.
Very much Truth in Television. Whether justified or not, a person is expected to do their job well, and a disproportionate amount of attention is almost always placed on any and all mistakes.
Another in-universe case from SVU- Elliot Stabler once admitted that he has fantasies of murdering child molesters, which is often brought up by people who question his credibility as a cop.
While The Shield star David Rees Snell (aka "Ronnie Gardocki") had a manly beard for the bulk of the series, it's not the beard that the actor is most remembered for facial hairwise; it's his magnificent Porn Stache that David Rees Snell had for the first two seasons of the show.
When Law & Order's Jack McCoy has an argument with one of his subordinates over questionable tactics, expect the phrase "You once hid a witness" to come up. In the episode "Under The Influence" (s8e11), McCoy hid an exculpatory witness from the defense, in order to maintain murder charges against a drunk driver who killed two pedestrians (a mother and daughter). (He later relents, but still faced sanctions for his actions).
Similarly, expect the fact that he slept with Claire Kincaid to pop up at least once a season.
Mike Cutter also liked to point out the time that he held a bunch of Russian gangsters without charge for weeks on end, and took it almost all the way to the Supreme Court.
Mike Logan got Put on a Bus for punching a politician. He features in a later TV Movie. Naturally, when Law & Order: Criminal Intent rolls around, he's gotten a reputation as a hothead, and carries around a clipping about the incident in his wallet. Though having a temper and an attitude was part of Logan's character from the start, it really didn't get thrown back in his face until CI.
During her time on General Hospital, whenever Winifred and Maxie were on the same screen, she would always remind her that Winnie was the one who put Spinelli in prison (as part of her job as an FBI agent). By the end of her run, she wasn't even an FBI agent anymore. She was removed from the show forother reasons.
Sam McCall is well known for kidnapping Jason's son Jake. This is brought up every time Jason and Samantha share screen time. Despite the fact that she didn't even kidnap the boy; she merely kept quiet about who did. Even better, she would later save the same child from a burning building (in a separate incident), but she's still known as the one who kidnapped Jake.
In-universe example: Agent Booth on Bones once, in a moment of personal stress, drew his weapon and fired two rounds into a robotic clown-head atop an ice cream truck. Several seasons later, after he'd completed counseling, got reinstated and received commendations for his work, it still gets brought up by folks from other government agencies when they want an excuse not to trust him with sensitive documents.
Similarly, Brennan finds herself constantly reminded that she once shot an unarmed man (it's okay, though: he was trying to set her on fire).
And he was trying to destroy evidence!
The Nanny's Nanny Fine will occasionally rub Mr. Sheffield's nose in his decision to pass on producing Cats for Broadway.
Andrew Lloyd Webber is the Always Someone Better for Mr. Sheffield, so Niles will rub his face in Webber's successes even more often than Fran. Also, a few seasons into the show, Maxwell said he loved Fran... and then took it back. She brings that up about every other episode.
Burt Newton's infamous line about Muhummad Ali at the Australian Logies: I like the boy. Meant without any malice at all, but he will be forever known (outside of Australia) as 'That racist guy who nearly got beat up by Muhummad Ali'.
Find yourself on a long-form reality gameshow? Don't say anything silly in the first week!
Shadow Moon is undoubtably the most popular Kamen Rider villains. And that's all he's remembered as. Even though in his canonical death in Kamen Rider Black RX, he had a last moment Heel-Face Turn, that's always forgotten in favor of how kickass he was as a villain. (This is even in-show: in Kamen Rider World, he was the Big Bad, and giant-sized for no good reason, and the first Kamen Rider Decade movie has an Alternate Universe Shadow Moon who curb-stomps two Riders with the power to destroy the world at the same time until Kamen Rider Double shows up and completely turns the tables. Not one word about getting Nobuhiko back to his family's side has ever come up in his non-Black or Black RX appearances.)
JAG: In the second season episode "Heroes", Harm fired off a sub-machine gun in the courtroom to demonstrate a point. This gets referenced two or three times a season for the rest of the show's run, usually in terms of "I can't believe he didn't get brig time for that", or "you should've seen him in the courtroom."
Many LOST fans tend to ignore Michael's more positive (or at least less negative) traits after his Moral Event Horizon moment in season 2 (i.e., murdering Ana Lucia and Libby). While the act was certainly indefensible (which makes this a partial case of Justified Trope), fans gloss over the fact that having your son kidnapped by strangers on a weird island doesn't exactly make a loving parent rational, nor did the fans acknowledge what he did AFTERWARD, which contradicts the assumption that he's an amoral, heartless bastard. This includes never ending guilt for doing the aforementioned act, which sparked numerous suicide attempts, and a last ditch effort to help the friends he betrayed on the island. Hell, even Hurley later forgave Michael for what he did, despite him killing Hurley's girlfriend Libby. Good luck finding fans who feel the same way Hurley did.
Jack is rather well-known for his frequent emotional outbursts (Jears) around the interwebs. In the actual show, he's a mostly-stoic character (for the first few seasons, anyway) who relies on logic and rarely tells people how he feels.
MSTings have a Running Gag dubbed "Crow Syndrome", where Crow (or another character) almost constantly makes sexually suggestive riffs and gets a First Name Ultimatum from the Team Dad. This seems to be based entirely on the episode Riding with Death, where everyone uses the film's trucking scenes as sex metaphors; Crow is just the one who takes it a hair too far and gets chewed out by Mike. Of course, usually he displays Ping Pong Na´vetÚ; compare to the episode where he puts together a presentation about how women don't exist, despite interacting with Pearl Forrester for years. In regards to MSTings, Crow Syndrome has become a Discredited Meme and is now viewed as something to be avoided.
Amusingly enough, most people tend to forget that in the Riding with Death episode, Mike himself makes a suggestive joke shortly after reprimanding Crow, who responds "And you think I'm bad?".
Prof. Bobo, the Simple Country Lawyer: So you blow yourself up a planet; does that make you a world-destroyer? Hmm? My momma, she burnt a brown betty one time, that make her a world-destroyer? I reckon not.
If you went strictly by MSTings, Tom Servo's head exploded every other episode. It only happened four times in ten seasons: Two of them in the disavowed Season 1, and none after Season 4.
Unless you also count The Movie, where poor Tom kept getting hit by death rays.
Red Dwarf gave us a justified in-universe use, then lampshaded it when Arnold Rimmer reads of the captain having described him as "constantly failing" the astronavigation exam:
Rimmer: "Constantly fails the exam? I'd hardly call 11 times "constantly." I mean, if you eat roast beef eleven times in your life, one would hardly say that person "constantly" eats roast beef, would you?
Lancelot from Merlin has a reputation among the fandom for being something of a dolt. This is distinctly odd considering he is one of only two characters to have deduced that Merlin has magic, and picks up on the sparks between Arthur and Guinevere before even they are fully aware of it. Yet so many times you'll see him described as "a bit dim", perhaps because he takes the Honour Before Reason trope Up to Eleven.
Likewise, Guinevere's reputation as a useless Distressed Damsel. She had a total of two episodes in five years dedicated to rescuing her, and both times she was extremely outmatched. Other times, she's a competent Badass Damsel who spends most of her time rescuing other people.
Some people will never forget the time Gwen Cooper of Torchwood confessed to her boyfriend that she'd cheated on him, demanded his forgiveness, and then retconned his memory so that he couldn't remember it. The fact that it's never been brought up since doesn't help, but that Gwen has married, had a child with, and remained faithful to that same boyfriend doesn't seem to win her any points either.
Jack? Yeah, he's that guy who dies once an episode, right? Now, not to say he doesn't die often, but saying he does so that frequently on the show is definitely stretching the truth. The Children of Earth miniseries is a special exception, but realise that most of Jack's team didn't even know he was immortal until the final episode of series 1. Prior to his constant state of suffocation and rebirth in "Exit Wounds" (which is another special example where he feasibly died millions of times while buried alive), he died nine times in seven episodes, plus twice in a flashback of one of those episodes; barely a quarter of the episodes that had aired up to that point. Following Children of Earth, there were only two episodes in Miracle Day where he was shown to die, due to his new state of being mortal for much of that series, and one of those was only in a flashback.
Owen's introduction, where he uses a spray on a woman that makes her instantly want to sleep with him, and then on her boyfriend when he objects. Yeah, throwing in Bi the Way doesn't really distract from the fact that our first impression of the guy is as a serial date rapist.
Though Tim Taylor of Home Improvement had had many, many, many accidents over the years, for some reason he never lived down that one time he glued his forehead to the table.
Except, maybe, sticking his tongue to a frozen hammer in the first Christmas Episode.
Ben Wyatt of Parks and Recreation won a brief measure of nationwide fame when he rode a wave of local anti-incumbent sentiment and got elected mayor of his town at the tender age of 18. He got some more fame when he bankrupted his town and got impeached after a month. Needless to say, he doesn't like being reminded of it, which any citizen with a search engine is happy to do.
Clare in Degrassi ranted out Eli at the Local Hangout, screaming "DID YOU FLIP A SWITCH AND ERASE ME FROM YOUR MEMORY, DID YOU EVER LOVE ME AT ALL?". Since then, Eli wrote it into the School Play, Connor and Mo used it to psych out an Opposing Sports Team, and Connor, KC and Adam put it on Twitter where at one point it was trending in Toronto.
Jake Martin: chicken connoisseur. Though he was only shown eating chicken in one episode, you'd think that he eats it for breakfast, lunch, and dinner.
In-universe in Glee, people are still making jabs about Tina putting vapo-rub on Blaine's chest while he was passed out.
Fandom will probably never forgive Finn for inadvertently causing Santana to be outed not only to the entire school, but the entire town. It just goes to show how selective fandom's memory can be, since Finn only snapped back after being bullied by Santana all day and had no idea that he was being filmed by someone who was out to get Sue Sylvester.
Scrubs: The fandom will never forget about Carla making JD move out of the apartment (that was his to begin with) once she and Turk got married.
People are never going to let The Dating Game live down the fact that they once let a registered sex offender who later turned out to be a serial killer not only get on the show, but win the episode he was in.
In-universe example in M*A*S*H, during an episode when Hot Lips demands a transfer from the 4077th, citing Hawkeye and Trapper's hijinks as one cause:
Hot Lips: I am not looking for a truce with these two shower tent peekers!
Trapper: Boy, you peek into one shower and you're labeled for life.
Dennis Franz showed his naked ass for a very small part of one episode of NYPD Blue. Ask a casual fan and you'd think the guy never wore pants.
The Walking Dead racked up a lot of criticism in its third season for the writers apparently having a rule that there could only be one black man on the show at a time, and any time a new one showed up you could count on the one already there being killed off. This was the result of some unfortunate timing in just two episodes, and also ignores Michonne (a black woman) being a major character.
An in-universe example from Sherlock. Despite only wearing a deerstalker cap once in an attempt to disguise his appearance, it is referred to constantly by the press as a signature item.
Jessica Brody from Homeland going on a racist tirade about how evil Islam is in the season two premiere, capped off by desecrating her husband's Quaran. The idea was apparently that her major concern was him keeping his conversion hidden from her for so long, but it certainly doesn't come off that way.
Daenerys in Game of Thrones actually did more things in season 2 than run around yelling "Where are my dragons?" It didn't even start until the season's second half. But that second half packed in so much of it, while everyone else was doing much more interesting things, that it really sticks in the mind.