Tony during the first season was more or less nothing much more than a typical Jerkass who generally found any excuse possible to go up against Jack with a lot of it being mere jealousy. To say fans hated him back then is an understatement, with some even wishing he'd been The Mole just as an excuse to kill him off or otherwise get rid of him. Around midway through the season he started transitioning to Jerk with a Heart of Gold status, as he ultimately showed sympathy to Jack's plight in season 1 in spite of their hostility and he proved himself to be one of the few agents that wound up being trustworthy for the whole season with things culminating in a well received scene where he saved Jack's wife from an assassin.
Beginning in season 4, the producers took great lengths to rescue Chloe from the pit of fan hatred (even earning the nickname "Jar-Jar" on one forum) that she'd fallen into after her début the previous season. She was given better makeup and wardrobe, they toned down her abrasive personality, and ultimately gave her a Crowning Moment Of Awesome — a scene where she mowed down terrorist assassins with an M-16. It also helped that her "I do things my way and don't listen to authority" attitude, which was directed at well-liked characters like Jack in season 3, were later redirected at unsympathetic characters. More simply put, she could still be a pain in the ass, but if she is a pain in the jerkass character's ass, driving the unsympathetic character up the wall, you love her for it. This was later lampshaded in the final episode of 24. In that episode, Jack actually said something along the lines of "When you first came to CTU, I never thought it was going to be you that was going to cover my back all those years." to Chloe during their final conversation. Chloe later has the final line of the series, "Shut it down" before the 24 countdown clock times out.
Kimberly used have loads of hatred for being the Damsel Scrappy but this changed as she grew up. When this happened is debatable, though - most would argue Day 7, when she apologises to Jack for how she acted after he came back from the dead, or Day 8, when she tells him to go back to CTU regardless of whatever plans he had with her, but some say Day 3, when she was working for CTU and managed to make herself useful more often than not. It also doesn't hurt that she Took a Level in Badass on Day 7, and was able to get out of a terrorist's grasp using nothing more than a pen.
For the first quarter of Day 6, Tom Lennox wasn't much more than the typical pain in the ass government employee, and just about everyone watching were quick to dub him one of the worst characters on the show as early as the season premiere. During the remainder of the season, his character was quickly established as one who in spite of some of his questionable tactics truly did care about the fate of the country and after seemingly crossing the Moral Event Horizon by agreeing to take part in an assassination attempt against the current President later proved himself by revealing he was attempting to expose the conspirators all along; ultimately he even went on to serve as the Morality Pet to a much more gung-ho extremist President later on. Despite the fact that the sixth season is consistently seen as one of the weakest, even its detractors generally agree that he was one of the few real bright spots in it.
Ryan Chappelle was probably the epitome of Obstructive Bureaucrat in the show's early seasons. Most of his screentime involved him getting in the ways of Jack and (especially) Tony, to the point where his reluctance to aid led to one of Jack's suspects dying from blood loss. However, after Stephen Saunders ordered the President to execute him (for fear that Chappelle discover something in Saunders' accounts), fans immediately felt sympathy for the man, particularly when he let Jack fulfill the deed with minimal resistance. it's now regarded as one of the most powerful moments of the show.
Larry Moss was basically an FBI version of Chappelle in Season 7, holding Jack's feet to the fire for his constant disregard for protocol. This didn't endear him to a good number of the fans initially, but he grew on many as the season wore on. It likely started when he proactively led the countercharge to take back the White House from General Juma's men (notably against the orders of the hesitant Vice-President), and amplified when he aided Jack and Tony in taking down Jonas Hodges. When Tony killed him later on in the season, quite a few fans despaired.
American Idol: Haley Reinhart managed to pull herself out on Season 10. At the beginning of the season she was criticized for her appearance adding more to her staying power than her voice. That all changed when she started to deliver stellar performances and gained a possibly insane fanbase.
Battlestar Galactica (Reimagined): Anders and Cally in the reimagined show. Anders was disliked for distracting Starbuck from Lee, but got his own characterization arc after realising he was a Cylon and turning out to be more important for several reasons than it first appeared. Meanwhile, even those who liked Lee at first went off him after too much smugness when he left the armed forces. Cally replaced Boomer in the chief's affections and seemed (realistically) tough and bad-tempered. She got her own episode though just before Tory offed her.
Bones: Could qualify as a Trope Codifier or Trope Maker with the number of characters that have been rescued from becoming The Scrappy, it's hard to know where to start. We could start with Zach Addy, who started off as an adult Cousin Oliver but as season one progressed, became a very strong character who lasted for 3 seasons before being Put on a Bus (this troper is positive that fans rioted after that). Then there was Dr. Saroyan, who started off as a taskmaster of a boss and a Jerk Ass, but after nearly getting Killed Off for Real midway through season two, her popularity soared. Then there's Dr. Sweets, who was just like Zach, but with more social skills. Fans didn't really like him until he offered valuable contributions to the Gormagon case. (Incidentally, this was about the same time Zack left.) Some of the Squinterns have become popular after their inauspicious debuts, like Dr. Edison, Vincent Nigel-Murray (who ended up being Killed Off for Real after 3 seasons) and Arastoo Vaziri. Oh, and lets not forget Caroline Julian, who has made her fair share of fans after her inauspicious debut. In short, no matter where you look, Bones is definitely a Trope Codifier for this.
Skyler was rescued in Season 5 and became a Jerkass Woobie as Walt's Heisenberg persona started to come into their married life.
Zig-Zagged with Marie. Marie was rescued in Seasons 3 and 4 when she takes care of Hank and tries to keep him from becoming depressed after a shootout leaves him bed-ridden. She slipped back into Scrappy territory in the first part of season 5, but has well and truly left the zone in the second part when she finds out about her brother-in-law's activities and her sister's knowledge of these activities, with a slap in "Buried" and a Wham Line in "Confessions.
Hank was rescued when he had a nervous breakdown over killing Tuco and the bombing in Juarez.
Buffy the Vampire Slayer: Wesley was probably The Scrappy, and seemed to remain under this shadow at first when transferring over to Angel. However, he quickly improved, and after a couple of seasons and a great deal of character development, he was one of the more fleshed out and well-rounded characters on the show.
Connor, after the Wolfram Hart Deal with the Devil that Angel made, came out a much nicer, sane, laid-back fellow after having new memories and a new upbringing put in. Even after he got his memories back, he still was much improved.
Dawn: her very existence was a Cosmic Retcon, and many felt that she was a whiny Damsel Scrappy. In the last season she loses some of her angst issues and becomes a sort of junior Watcher, helping the rest of the team in research.
Tara was originally despised for replacing Oz as Willow's love interest, but over time the fans came to love her and then revolted when she went off the show and was followed by another Replacement Scrappy, Kennedy, who was despised through the rest of the series and into the comics despite several attempts to make her a character most fans would consider really cool were it anyone else.
Days of Our Lives: Chelsea was an extremely hated character when she first appeared. She was an arrogant Jerk Ass, rude to everyone around her (even her best friend Abby), responsible for killing her little brother, and tried to break up Bo and Hope's marriage. When she started dating Hollywood Nerd Nick, she became a much better person. She went from being the most hated character on the show to one of the more popular characters.
Doctor Who: When Donna Noble made her first appearance in the 2006 Christmas Special, many fans were put off by the shrill, abrasive nature of the character (Given that she was played by comedian Catherine Tate, at least the British fans knew what to expect). The announcement that she would be a regular member of the cast during the 2008 season was met with mixed reaction at best. A few episodes into the season, however, she had undergone deliberate Character Development, and about half of the fandom had warmed up to her and even praised the "shrill, abrasive nature" that earned Donna so much hate in her debut. As of the end of Series Four, certain fans even considered her if not the Best Companion Ever, then at least the best modern companion.
It was partly helped by circumstances. When Donna first appeared, that "shrill, abrasive nature" was levelled against the Doctor himself (including slapping him.) This isn't a smart thing to do around Whovians. By the time the show came around, that nature was being pointed toward the actual enemies of the series, so she probably started getting liked more when she started hating the right people.
One of the more interesting things about this example was that the other characters acknowledged that this had happened and in the end her fate worse than death was to be returned to her Scrappy-state. Some of the fans that once disliked her were instead upset that such an anti-climactic fate had befallen her.
In many ways this also extended to Catherine herself, since before her return to Who she was something of a victim of typecasting in roles which had a limited appeal. Donna started out as such a character, then changed incredibly.
Tate also picked up the Doctor Who Magazine's Greatest Contribution Award for 2008 for her performance. With twice as many votes as her nearest competitor, Russell T Davies, who had definitively proven Doctor Who was back and popular by leading it to its first #1 at the top of the weekly viewing figures in its history. If that's not proof how well the character turned around, what on Earth is?
Big Finishaudio appearances have considerably improved the reputation of the Sixth Doctor among the fan-base (they got rid of the hideous coat and gave him a blue one, for a start). To display just how much he has been rescued, it is now the case that the man who was once almost unanimously agreed by many to be the worst Doctor, now rates very highly on fan-polls, and is almost unanimously agreed by many to be the best Doctor of the Big Finish audios.
The Sixth Doctor's companions managed to get some boosts to their reputations as well. Peri isn't as abrasive as she was in the TV stories (and her accent slips less too) and Mel seems to have been updated into a Badass Bookworm with occasional moments of both Action Girl and Deadpan Snarker. And she doesn't scream as often either.
Similarly, the Seventh Doctor started his tenure as an annoying, over-the-top pratfalling clown with a tendency towards irritating mixed metaphors, who irritated many of the viewers. After his first season didn't go down that well, the production team decided to sober him up, taking away his more annoying quirks and adding a more mysterious, brooding depth to him. This met with widespread fan approval; however, it was arguably too late to save the series, which was cancelled after his third season. This Character Development continued in the Doctor Who New Adventures novels, which greatly increased the character's more sober, introspective Anti-Hero nature to widespread approval (although it's also often argued that, at worst, the novels had a tendency to take it too far with the Darker and Edgier stuff, and sometimes wobbled into making the character an unlikeable Knight Templar bastard). His Big Finish Doctor Who episodes largely take place between TV show episodes, and somehow, the writers took his first appearances as a bumbling goof and used that characterisation in two of the most well-loved audio episodes ever: "Bang-Bang-A-Boom!" and "Unregenerate!".
Another notable example is the Eleventh Doctor, for some. At the time of Matt Smith's casting, many fans cried out in panic that Ten was no longer going to hang around and in extreme cases threatened to stop watching the show entirely. This meant that all of Series 31/5/Fnarg was pretty much spent attempting to rescue the new incarnation for said section of the fandom. Haters that are left are generally Rose/10 Shippers or those who are too stubborn to admit that Matt Smith might be a decent Doctor after all. Granted that happens generally when they switch actors, but the Tennant-Smith switch made quite the fuss.
That one's probably debatable; it could be said that Smith received the same reaction as every actor playing the Doctor has received following the previous one. It makes more sense for it to be down to the fans, convinced that They Changed It, Now It Sucks and the more extreme of them going for the more intense "Ruined forever!" reaction. So in a sense, Matt Smith's Doctor didn't need rescuing from the Scrappy Heap because he was never a Scrappy to begin with.
It's worth noting that critical reaction to his performance was highly positive even immediately after his first episode. If he was ever a Scrappy, he was a very short-lived one, as the only footage of him as the Doctor previously was the few moments at the end of Ten's regeneration; some fans got a bad first impression due to the Mood Whiplash, and dreaded over-use of the Catch Phrase "Geronimo" (which didn't end up being said that much).
The most egregious Doctor Who example, Adric, seemed to manage this in the 2008 audio The Boy that Time Forgot, a tribute to the character and his status within the show's history, in which he is played by 'an actor' (Andrew Sachs) as opposed to Matthew Waterhouse; before that, his larger, more active role and Heroic Sacrifice in his final story went some way to redeem him, although that falls more under Alas, Poor Scrappy.
Farscape: The last two seasons had a couple examples of potential Scrappies who were rescued from the heap. Jool was initially a very annoying, shrill, and obnoxious character who underwent somewhat drastic character development over the course of the 3rd season and her brief appearance in the 4th, though for reasons unknown she devolved into a sex-starved Xena clone in the miniseries. The changeover appeared to occur around the time she began to be given duties approximating that of medical officer.
Sikozu, who joined in the final season, was initially dismissed as a Dropped a Bridge on Him - style replacement for Jool (even down to the red hair and obnoxious attitude), but was almost instantly rescued from the Scrappy Heap when she finally appeared on screen, most certainly around the time she let her hair down later in the first episode of Season 4.
Noranti, the crazy old woman added in the Season 3 finale episode, was most definitely a Scrappy at first, until hidden layers to the character were revealed, and the writers scaled back the "crazy old woman" in favor of making her closer in spirit (if not necessarily in looks and behavior) to Zhaan.
Glee: Used on purpose with Quinn. Quinn began as a shallow, nasty, Ice QueenAlpha Bitch who was just a Romantic False Lead for Finn — the Official Couple being Finn and Rachel. Quinn was also a spy for the Big Bad, Sue. She was so unlikable that you didn't feel sorry for her when she got pregnant due to this making her a Straw Hypocrite (she's president of the celibacy club) and got the baby while cheating on her boyfriend with his best friend. Over time, however, she turned into one of the most sympathetic characters on the series, turning into a Fallen Princess. Some people still hate her but even those who despise her admit to feeling sorry for her during episode 10 when a crying and desperate Quinn begs for her bigoted parents to forgive her only for them to promptly kick her out of the house.
Sadly, Quinn's been arguably pulled back in and out of the Scrappy Heap in the second and third seasons, though many of the fans blame the writers' attitude toward the character (as a dumping ground for random bad storylines) rather than hating Quinn herself.
Grey's Anatomy: Subverted Trope by Izzie Stevens, who earned a reprieve from the Scrappy Heap in her cancer storyline, which actually saw her become a pretty sympathetic character. However, a combination of the storyline dragging on without resolution and actress Katherine Heigl's extremely obnoxious real-life behavior have catapulted her back onto the Scrappy Heap in spectacular fashion, probably for all time.
Hannibal did this with Frederick Chilton, of all people. In the first season, he was portrayed as little more than an incompetent Smug Snake psychiatrist. In the second season, when Will Graham came under his "care," Chilton seemed to be gearing up to be even more of an epic Jerkass than before. Which... he still more or less was, but the showrunners took care to spotlight the more hilarious aspects of Raúl Esparza's performance, leading Chilton into Love to Hate territory. Then he got ahold of the Smart Ball and became an unlikely (albeit self-serving) ally of Will's, at a time when most of the rest of the cast was still treating Will as delusional about Hannibal. And then they basically dedicated an entire episode (Futamono) to Chilton being an Obnoxious Snarker about Hannibal Lecter's culinary interests, at which point he was officially rescued. By the time Hannibal sprang the trap he'd set for Chilton all along, fandom exploded with theories on how Chilton could still be alive.
How about Hiro? Popular in volumes 1 and then deeply annoying in 3 and 4 due to his development from the first two seasons being rejected in favor of him being a childish moron again. Then redeemed in 5 due to his cancer and Charlie arcs making him more serious again and him no longer appearing in every episode.
Dr. Cameron was initially hard to like, but the fanbase seems to have warmed up to her (especially in comparison to her replacement on House's team, Thirteen). It was difficult to like Cameron because she usually became emotionally involved with the patients and would usually disagree with House and sometimes the rest of the team simply because whatever House had planned was immoralnote The nature of the show makes any character who adheres to medical ethics look like an absolute moron. She was probably planned to be sort a straight man or voice of reason, but due to the strong personalities of the other characters, especially House, she became more like a preachy, annoying character who made stupid mistakes that sometimes did more harm than good for the sake of her morals. After being replaced by the new team, she reappears occasionally but is much less irritating.
Chase was similarly rescued - on House's team, he was an annoying suck-up with the moral fiber of soggy newspaper who had a crush on Cameron. After he got fired? Likable, funny, doesn't give a damn what House thinks, and in a stable and sweet relationship with Cameron. Rescued from the scrappy heap and Took a Level in Badass.
Thirteen herself got rescued when, after actress Olivia Wilde sat out much of a mediocre Season 7 to make movies, came back in an episode that many believed to be the best of the season, with several months of offscreen character development due to being in prison for euthanizing her brother, who was dying of advanced stage Huntington's disease.
House of Anubis: Joy Mercer was originally the show's scrappy, as she was in love with one half of the show's most popular pairing. Later she did do some pretty bad things, and earned the title for real even among other characters in the show- however, in the end of season 2 she took a lightning bolt for Fabian and nearly died in the process. Now in season 3 Joy gave herself a new look, a new attitude, a new crush, and has become one of the most liked characters on the show.
Jericho: Emily became The Scrappy very quickly, and remained so for all of Season One. Although she was now in a relationship with the hero, Season Two still made a good effort to rectify this, by giving her approximately two minutes of screentime an episode.
Joan of Arcadia: Judith from the second season started out as alternating between being annoying, a Jerk Ass, dangerously unstable, and seeming like she was either wanting to steal Adam from Joan or Joan from Adam. But as she got to know the other characters better and they got to know her, she calmed down a bit and revealed a genuinely sweet side to herself, culminating in the perfect date between Adam and Joan which she largely orchestrated single-handedly. And then, in a Wham Episode moment, she was murdered out of the blue.
LOST: While it may have been planned from the beginning, Jin became a far more sympathetic and likable character after the episode "In Translation", and especially in season 2.
While arguably more developed than Jack, Kate didn't really grow with most fans until either season 5 or 6 after she became a surrogate to Aaron and her love triangle with Jack and Sawyer lost focus.
However, for many fans, she never quite got rescued, even though the writers obviously tried; even in season 6 she was probably the most hated character in the show (a poll on a popular fan site showed that up to 50% of voters wanted her to die).
Invoked with Shannon, who was initially written as unlikable, becoming gradually less useless and annoying, culminating in her flashback episode, "Abandoned", which rescued her in the eyes of most fans and then killed her off. The writers were going to try the same tactic with Nikki and Paulo, but backlash against them was much stronger, and they were written out before they got a chance.
The L Word: Jenny was both The Scrappy and a Creator's Pet for most fans, but, when the writers caught onto just how disliked her character was, and brought her back as an all out unsympathetic, incompetent Jerk Ass director in Season Five, it actually boosted her popularity. Mia Kirshner's utterly hilarious performance with the new material made her a joy to watch. Ironically, having her go crazy - and acknowledging her craziness within the show - made fans care about her a lot more than any of the previous efforts to redeem or Woobify her in earlier seasons.
Mystery Science Theater 3000: When first introduced into the show, Pearl was rather unpopular among fans for being a smug Villain Sue mother of the established bad guy. Once she became the main bad guy however, her character was shifted until she became more of a Distaff Counterpart of the original Doctor Forrester.
NCIS: Ziva started out as a Replacement Scrappy for Kate. The writers attempted to rescue her by giving her a series of awesome talents and ninja skills and letting her perform a couple of selfless acts and save the day a few times, the most significant of which was when she helped Gibbs recover his memory in the third season finale, and it generally seemed to work. Ziva is now a pretty popular character and half of most of the fandom's OTP.
The fact that she's now been in eight seasons and Kate was only in two probably isn't hurting her popularity either.
Nikita: Jaden started out as a fellow trainee alongside Alex, who took a disliking to her for no apparent reason and acted like a school bully. Then comes the episode "Girl's Best Friend," where she proves willing to work with Alex for the good of a mission, gets some Character Development with her story about killing an abusive boyfriend, and finally is set up to be a full-fledged Evil Counterpart to Alex rather than just a whiny bitch. Unfortunately, she's killed off in the next episode.
The Office (US): Andy Bernard was introduced as a thoroughly dislikable sycophant with rage issues. While he's continued to be portrayed as a comically awkward character, his portrayal became more sympathetic during his engagement to Angela (who cheated on him and was otherwise emotionally abusive) and by the time he ended up a main character in the most recent series he was one of the most likable characters on the show.
Once Upon a Time: Though they've only been in two episodes, Cinderella and her Prince didn't impress anyone the first time around, mainly due to their idiotic behaviour and rather wooden acting. The next time they appear, the actors seem to have taken a few acting lessons, and manage to deliver an extremely sweet scene concerning a proposal at a busy restaurant. Perhaps they hold the record for how quickly two characters manage to redeem themselves.
Andy is a particularly good example. In the show's first season, which is 'very' well known for its prominent Early-Installment Weirdness, his entire purpose was to be Rashida Jones' terrible asshole idiotic selfish boyfriend whose stupidity was almost a superpower. In the second season, his asshole-ish properties were downplayed and eventually stripped completely. After the show's retooling, he quickly became a wildly sympathetic complete idiot who generally ends up doing the right thing of his own accord, because he's not a bad person, just a very stupid person. Even when he does something mean to someone, it's almost always because of ignorance. Basically he's a giant adorable baby now.
Person of Interest: Detective Joss Carter was originally an underdeveloped By-the-Book Cop who was concerned with chasing the main characters and brining them to justice. However, over time, she became one of the most beloved and respected characters on the show who in her final two episodes slipped into the role of The Chessmaster so effortlessly, almost all remaining critics and fans who used to find her the weakest aspect on the series were won over. Judging by the reaction to her demise in the Season 3 episode "The Crossing", many consider her The Heart of the show, the one who humanised the other amoral protagonists and brought the best out of all of them.
Power Rangers Lightspeed Rescue: Carter Grayson was originally considered a bland, uninteresting character. However, over time, he has become a Memetic Badass. The episode "Yesterday Again" showed that He Really Can Act. Also in comparison to the past rangers in the "Forever Red" episode he does come across as too reliant on his weapons, but few other rangers had the same "go in guns ablazing" mentality. So Carter doesn't use the typical strategies other Rangers use but instead just shoots at the bad guys or tries to run them over with the rescue jeep.
Ricardo "Rick" Medina is an interesting case. Originally playing the hated Cole in Wild Force, he was almost completely forgotten. Then, when Saban announced that Medina would be returning in Samurai, the fans almost went irate. This all changed, however, when his character Deker showed up. (That, and Medina himself was less of a jerk in Real Life as well.)
Blake Foster (Justin from Power Rangers Turbo) is another odd case. Justin was hated for being an annoying brat. However, ever since he finished puberty, fans are much more loving of Foster, especially since he has expressed his appreciation of the fandom. Justin himself was redeemed fairly well in the Power Rangers in SpaceReunion Show appearance, where he comes to save the day.
For one with a much faster turnaround, there's Alpha 6, also from Turbo. He was introduced as Alpha 5's Totally Radical, slightly ruder replacement. In the changeover to Power Rangers in Space, 6 had some emergency repairs that removed those traits and made him a straight Expy of 5, making him much more acceptable.
Back in the franchise's early years, before they began their Discard and Draw approach each season, any time an actor left, their replacement was hated on sight. However, given time and a little Character Development, most are able to shed their Suspiciously Similar Substitute origins and become well-liked on their own merits. Rocky, Adam, Kat, Tanya, TJ and Cassie. The only one who didn't get this was Aisha, who had the unfortunate problem of spending most of her tenure Out of Focus (though her actress is a fan favorite, at least.)
Despite being one of the scrappiest of scrappy characters on television, Danny manages to pull this off in one of the most spectacular ways possible. Miles is knocked out by a chopper missile during an attack on a rebel HQ while attempting to take it out. Danny makes a run for the rocket launcher, and successfully destroys the chopper carrying the amplifier, resulting in the other chopper losing power and dropping as well. Unfortunately for Danny, he is torn to shreds by stray machine gun bullets not ten seconds after. To reiterate, Danny not only saved everyone at the rebel HQ, but destroyed Monroe's only power amplifier, so possibly thousands of other lives as well. Later, according to Jason, the story of Danny's last stand has spread and inspired many other people—possibly other defectors from the militia—to join the rebels, reinvigorating their cause after it was nearly destroyed.
Charlie is (albeit slowly) starting to get rescued too. She's not whining nearly as much as she used to, and she's finally starting to live up to her title of being the Action Girl for the series, alongside Nora.
Robin Hood and Downton Abbey: An interesting meta-example happened across these two shows. In Downton Abbey, Joanne Froggatt plays the plucky and down-to-earth Anna, a character that is immensely popular within the fanbase. This is in stark contrast to her stint as the infinitely irritating Kate on Robin Hood, (which directly preceded her role in Downton) where she was completely and utterly loathed by viewers. It just goes to show how much an actor's likability depends on good writing. Even Jo herself seems to be aware of it, considering Robin Hood doesn't appear on her resume.
Royal Pains: Evan was The Scrappy for most of the show by being the annoying younger brother to Hank. Evan's main concern seemed to be expanding Hank Med against Hank's protests. Though since he started dating Paige, he's become a much more likable person, improving his character immensely.
Scandal: Quinn Perkins started out as being hated, due to being the Naïve Newcomer and generally sticking out like a sore thumb. Season 2 fixed this by revealing that she is actually Lindsay Dwyer, a woman who is wanted for murdering 7 people. She did no such thing, but there are corrupt powerful people out to make her the patsy. Quinn is now considered The Woobie. In addition, she is receiving training from Huck, and is turning into a female version of him.
Shameless: Kelly Ball. Started life off as Kev's annoying sister. After annoying out of everyone in one episode per series (2-4), she was promoted to the main cast as Shane Maguire's boyfriend and has shed her previous image.
Skins: Series 3 introduced Cook, who was initially regarded by fans as an unlikeable hedonistic thug with no redeeming features. Series 4, while not removing the thuggish image, redeemed Cook in the eyes of many fans by portraying him as a loyal friend and a caring older brother.
Star Trek: The Next Generation: Then there's Wesley Crusher himself. During the run, Wesley had his Pet-ness toned down at least somewhat as the show went on. Even Wil Wheaton himself relates that he insisted the writers bring the character back down to more realistic levels or he'd quit. This wasn't enough to save him, though, and he ended up being Put on a Bus. Nobody missed him.
All of Wesley's reappearances were better than almost any episode he had a major role in during the first four seasons. One of them ("The First Duty") is generally considered to be one of the show's best episodes.
"The Game" is another decent example. A popular episode, Wesley manages to serve as a non-irritating protagonist (aided by Ashley Judd) and isn't even the one to save the day.
Star Trek: Deep Space Nine's Nog annoyed viewers to no end in the early seasons of the show, but he went on to become one of its most sympathetic characters.
Jake and Rom as well - to a certain extent, anyway. Jake grew up along with Nog and Rom became more likable when it turned out that he could do things besides screw up.
Julian Bashir was intended to be a Scrappy eventually rescued by humanizing flaws.
Survivor: Jerri Manthey. Everyone hated "Man-Eater Manthey" after her appearance in Survivor: Australia back when Survivor was new and it was the next greatest thing. She received similar treatment after appearing on Survivor: All Stars, when she was booed out of the reunion by the audience. Her third time on the show in Survivor: Heroes Vs. Villains, she toned her self down, and was put next to larger than life characters like Coach and Russell Hantz who many fans did not like. She turned into the Plucky Girl and became a fan favorite near the end before being voted off at the final four. She received a loud applause at the reunion, in contrast to her reception several years prior. Her lovely black Army hat didn't hurt, either.
In the same season (heroes vs. villains), Coach actually became this. In Tocantins he was a Cloud Cuckoolander who was a Creator's Pet hands down. However, in Heroes vs. Villains, he seemed to focus more on the game itself, begun to perform well at challenges and pull his weight in the game.
Hilariously enough, Coach and Jerri were actually allies during that season.
True Blood: Jessica quickly shed her Scrappy status after she started seeing Hoyt, a relationship that many fans now consider the most realistic and relatable on the show..
Same goes for Tara, who despite Rutina Wesley's strong acting performance was deemed "too angsty" and not relatable enough. In season 5, she becomes a vampire, with Pam as her maker, had several awesome moments and of course The Big Damn Kiss with Pam.
Eric's vampire sister Nora was partially rescued from the heap halfway through season 5 after she lost her insane religious devotion. Seeing her cry for a vision of her maker Godric being obliterated by Lilith made her downright sympathetic.
Matt as well in Season 3 after he takes on the role of Only Sane Man.
Vanessa was universally hated among the Gossip Girl fandom to the point that they eventually wrote her out. Only, in the last season she appeared the writers seemed to think the fans would enjoy seeing her being treated like crap and went with that full-force. The result was that people began to sympathize with her and turned their hatred towards Dan who was the character who treated her the worst (while sanctimoniously telling her that she was an awful person even though he was the one treating her badly). It didn't help that Dan was already borderline Scrappy (and became the Wesley full force once Vanessa was gone).
Delia in In Plain Sight. She starts out extremely annoying, with Mary (the protagonist) very much not a fan. The second season of the final season, "Four Marshalls and a Baby", seems largely designed just to make Delia more likable. She proves herself to be very adept dealing with baby Nora (Mary's daughter), owns a neo-Nazi with just a few lines of dialogue and an icy stare, and talks down an unstable witness. By the end of the episode, Mary is won over...and so is the audience.