Agent Pendergast has Pendergast/Corrie, Pendergast/Diogenes, Pendergast/D'Agosta, Pendergast/Constance...
Alice and the Mad Hatter in Alice in Wonderland. The original book is for children, but some adaptations and sequels gave their relationship romantic traits. Naturally, Alice/Hatter is now the most desired ship.
And at Disneyland, two of the costumed actors who play them are actually in a relationship, and therefore the Disney incarnations have a pretty big following on sites like Deviantart as well.
One of the oldest examples of this is shown in the Portuguese/Spanish (its authorship is disputed) medieval romance Amadis Of Gaul. Except that in this case it's not so much Fan-Preferred Couple as King Preferred Couple. To further elaborate: the hero Amadis is inextricably faithful to the princess Oriana. During one of his quests, however, one lady Briolanja falls desperately in love with him, and puts him in jail because he refuses to go to bed with her. Apparently the king of Portugal at the time, D. Afonso, really liked Briolanja and wanted her to have a happy ending, so he himself ordered the book to be changed, and have Amadis get hitched with Briolanja (with Oriana's permission, otherwise he'd die... it's a long story). This is retconned in later editions of the book, where the "author"/compiler himself narrates this version of the story, but immediately says afterwards something along the lines of "but this is just the king's whim and not what really happened". And then goes on with the "original" version. It's bizarre.
Artemis/Holly in Artemis Fowl has been the Fan Preferred Couple since the very first book. Even when Holly HATED Artemis.
Which takes Mayfly–December Romance to an extreme because not only is Holly very long-lived, but Artemis began the series as a prepubescent child.
The Casteel Series: a lot of readers preferred Heaven to be with Troy instead of Logan, even after Troy was revealed to be Heaven's uncle. This is because, aside from incest not exactly being rare in Andrews' works, Troy was the Troubled, but Cute guy female readers swooned over who actually turned out to be more thoughtful and caring towards Heaven than Logan was, making readers wish that he didn't have to be related to Heaven.
An odd example with Mary Renault's The Charioteer. Laurie and Ralph end up together, so they're actually canon. However, this was not intended as an unmitigated happy ending — almost none of Renault's endings are. Many fans, particularly the straight female fanbase though also some in the gay male fanbase, think that Ralph was supposed to be the right guy for Laurie all along, when in fact Renault wrote in a letter that at the end Ralph was "being taken on out of pity" by Laurie.
With Samuel Richardson's epistolary novel, Clarissa, readers wrote to the author before the last volume was released, asking him to give the story a happy ending involving a wedding between the heroine, Clarissa, and the villain, Lovelace. Readers liked Lovelace despite his actions, which included raping the main character, and thought Clarissa was prudish and a tease. Richardson not only refused to change his plans for a tragic ending, he also rewrote parts of the novel for later editions adding 200 pages and making Lovelace even more of a villain. Some readers were so against the tragic ending of the story that they wrote their own versions. Two examples of these shippers are Lady Bradshaigh and Lady Elizabeth Echlin. While Bradshaigh just rewrote the ending, giving Clarissa and Lovelace a wedding on his deathbed, Echlin rewrote her own version of the entire novel.
Edmond Dantes/Mercedes Herrera in The Count of Monte Cristo. They actually were canon for a while, but events occurred to split them apart. Many fans do prefer Edmond/Mercedes to Edmond/Haydee, including the makers of the 2002 film.
In Darkest Powers, Derek and Chloe were the fan preferred couple from the very first book, despite Derek's brother flirting with Chloe and Chloe tentatively reciprocating.
To elaborate, both of them are canonically Bi the Way, and the Doctor has No Sense of Personal Space and has kissed Fitz more than once just because he was happy to see him. Fitz has on a few occasions admitted to having a crush on the Doctor, and eventually flat out admits he's in love with him. While the BBC didn't really allow anyone to explicitly portray the Doctor as in love with another man, many of the writers slipped in hints whenever possible, and towards the end of the series's run, most of the recurring writers had given up trying to get crap past the radar because it had become patently obvious that the radar didn't care.
In The Dresden Files the majority of the fanbase shipped Harry/Murphy (who are best friends on the edge of Relationship Upgrade) over the official Harry/Luccio. It turns out that Luccio was brainwashed into dating Harry, and they break up. And since Changes there's been more official Ship Tease for Harry/Murphy.
Weirdly, the fanbase has a large slice of Yaoi Fangirls (Fanboys?) who ship Harry, the Big Good, with Marcone, the Affably Evil mob boss. This is strange because Harry's narration has a lot of Male Gaze, and he is Ambiguously Bi at most. At any rate, this one isn't about to become canon.
Many - possibly even a majority - of fans of the Earth's Children series wished Ayla had stayed with charismatic Ranec from The Mammoth Hunters. Indeed it seemed like Ayla only chose to go back to Jondolar (who was being a petulant ass throughout the book) because Ranec is black. That... or Ayla's a size queen.
There's also those who think Baby ate the wrong brother ship Ayla/Thonolan, Jondolar's brother, who was dead before Ayla and Jondolar ever met.
A good chunk of the Emily of New Moon fandom favours Emily/Dean over Emily/Teddy, as even fans of Emily/Teddy find him a painfully dull, flat character with all the personality of a paperclip.
In Georgina Kincaid, Georgina/Seth is often ditched in favor of Georgina/Carter, Georgina/Jerome, and Georgina/Roman.
The Grisha Trilogy has a rare case of two involving the same person. Alina/The Darkling and Alina/Nikolai are both more popular than Alina/Mal. This seems to stem from a large number of fans finding Mal, Alina's main love interest, boring at best and borderline abusive at worst. A good number also feel Alina simply has more chemistry with the others.
Judging by fanfiction.net, Draco/Hermione is the most popular pairing in the fandom involving either of these two characters. With the amount of stories it gets, it doesn't just beat out all alternative pairings involving one of these characters, it beats the canon pairings, too.
Draco/Harry completely out-wipes all other pairings on archiveofourown.org. As of February 2017, Draco/Harry fanfiction has over 20,000 stories, compared to the next popular pairing on the same sight (Sirius/Remus), with just under 10,000. And canon pairing Ron/Hermione has just under 6,000.
Neville/Luna is very popular. Neville and Luna don't have any canon pairings listed in the main story and Luna's partner is only revealed in interviews with the author as Rolf Scamander. This has led to the fandom term "Rolfing", a derisory phrase applied to canon characters, or author-invented characters, that exist only to get in the way of a fan's preferred couple.
Lupin/Sirius vastly overshadows the canon Lupin/Tonks pairing. It is so popular, that many fans actually thought the two characters were canonically together, including the two actors.
Although Scorpius/Rose seemed the most popular of the next generation character ships for a time, it was soon VASTLY overshadowed by Scorpius/Albus after the release of Harry Potter and the Cursed Child, which contained so much dialogue of affection and care for one another that many readers genuinely thought it was headed towards a romantic relationship, with many popular websites such as the Guardian or Vox accusing the book of queerbaiting.
And of course, on the other side of the war, there's always Achilles/Patroclus.
Many readers of Ivanhoe have expressed a preference for Wilfred/Rebecca over Wilfred/Rowena. This was probably intentional. Rebecca is clearly the heroine of the book (more so than Ivanhoe as the hero); the reason they didn't end up together was that the Canon couple had been in love for a long time and Rebecca would never marry a Christian. It's probably a rare case of the author himself being one of those who preferred an alternate couple!
Not just probably it is. Scott said so in the introduction. Scott also said it would have been an over-cheap and unrealistic morality lesson to have characters rewarded in this life for their good deeds with a successful spur-of-the-moment romance.
Pre-television example in Little Women due to it being autobiographical: Jo and Laurie become best friends and understand each other better than anyone else, but when the Relationship Upgrade moment comes, Jo uses the old "Like Brother and Sister" excuse, and at some point, she wants Laurie to marry her little sister Beth. The two end up marrying different people (Jo marries her Big Brother Mentor Fritz, Laurie marries Jo's other sister Amy since Beth dies)... and continue to remain best friends and confidants.
Alcott had her own reasons for this. Her father had high ideas and forced the family to live in a commune and try and be self-sufficient while he played philosopher all day. In order to prevent her family from starving, she wrote Little Women which actually was only the first half of Little Women as we know it today. It was a great success and so her publishers pushed her for another. Fans particularly pressured her to find husbands for the little women, and to get Jo and Laurie together. However the book was very much based on Alcott's own family, with Jo based on herself. Alcott wanted Jo, like herself, to remain an independent unmarried woman who earned her living through writing. However, with her family still starving, she had no other choice but to write the sequel Good Wives, but refused to let Laurie and Jo get together.
Meanwhile in the continuing sagas of Literature/Little Men and especially the latter Literature/Jo's Boys where the cast is all grown up - some fans still prefer Nan/Tommy over Tommy/Dora even though the former ship was sunk over and over by the author. Dan dying in the epilogue of Jo's Boys outraged everyone, especially when some wanted to see Dan/Bess happen. Other fans also liked the thought of Dan/Nat or Dan/Nan.
In The Lord of the Rings, many readers were taken by surprise by Sam's sudden interest in otherwise barely mentioned Rosie, after a trilogy of obsessive devotion to "Master" Frodo. Compare the amount of Rose/Sam fanworks to Frodo/Sam.
Aragorn/Eowyn fans are a good deal more numerous than Aragorn/Arwen fans. It mostly stems from Arwen's near-total lack of pagetime - even movie fans often lament how Arwen's not half as interesting, even with all the Adaptation Expansion she got. Most of the development for Aragorn/Arwen is off-page, to say nothing of Eowyn/Faramir.
Quite a few Janeites wish Mansfield Park had ended with Fanny/Henry Crawford and Edmund/Mary Crawford instead of Fanny/Edmund, despite the fact that Austen always strives to show that true love arises from similarity of character and that bad boys will not change for a good girl.
A large number of fans think that Marius chose wrong in picking sheltered ingenue Cosette over street-smart Stalker with a Crush Eponine. Never mind that he didn't know the latter loved him until right before she dies in the book, and never in the stage show.
Enjolras/Grantaire also have a huge fanbase.
In Agatha Christie's Miss Marple book 4.50 from Paddington, Lucy Eylesbarrow had two suitors, Cedric Crackenthorpe and Bryan Eastley. In her "Secret Notebooks", Christie reveals that Lucy would end up with the former, even though many fans find him the more unlikable of the two options. The one that most readers really want Lucy to marry is actually Inspector Dermot Craddock, whom she interacted with once or twice.
Despite her having two canon lovers, the majority of the fandom of The Sookie Stackhouse Mysteries are rooting for Sookie to end up with either her boss, Sam the shape-shifter, or Bill's "boss", Eric Northman.
Eric Northman became the latest canon relationship. (They'd had sex before too.)
That relationship eventually falls apart as well. By the epilogue, Sookie is happily married to Sam.
And then came the "Hand of Thrawn" duology, which not only made Mara/Luke canon, but also seemed like Zahn's attempt at a Fix Fic of a few of the other more controversial decisions of the EU up to that point.
Despite (because of?) the lengths gone to establish Ashe as Rhapsody's soul mate, many Symphony of Ages fans prefer to pair her with Achmed. Possibly subverted in that Achmed has decided to wait to get with Rhapsody until after Ashe dies making the couple borderline canon after all.
Averted in Song of the Lioness with Alanna's childhood friend George Cooper, who Pierce decided was better for Alanna after she sunk the ship of Alanna/Jonathan in The Woman Who Rides Like A Man.
Played straight with Beka and Rosto in Provost's Dog, as she ends up marrying Farmer, who didn't appear in the first two books.
Many fans of Twilight think that Bella would have been better off with Jacob than Edward. A lot of reviewers of the film adaptation of New Moon agreed with them, saying that Bella had more chemistry and believable development with Jacob than she did with Edward.
Even though the canon pairing is Gregor/Luxa, many The Underland Chronicles fans seem to prefer Gregor/Howard for some reason.
Warcraft: The Last Guardian has Garona and Khadgar; so much so that when Garona was first announced to have a son with magical potential, many fans assumed the father would be Khadgar.
Warrior Cats: Jayfeather/Briarlight. The fact that both of them are disabled yet determined to help their Clan, along with how she moved in with him after the accident that paralyzed her, quickly caused them to become one of the most popular pairings in the fandom. The fact that Ship Tease is thrown their way like water on a fire doesn't help (or does, depending on your point of view).
Lionblaze was commonly paired with Icecloud and Hazetail in fanworks, the former because she barely got any screen-time, and the latter because it was hinted that he had a crush on her. His Relationship Upgrade with Cinderheart did nothing to stop the fans.
The Woman in White has Walter and Marian. His close, equal friendship with her and how tough and brilliant she is despite her ugliness make the couple much more interesting than his romance with the beautiful but boring and helpless Damsel in Distress Laura.