The sequel Love Never Diesderails Raoul into an alcoholic, emotionally neglectful husband just to raise the possibility that the Phantom and Christine can be an official couple. It's revealed that she always loved the Phantom more all along, and they are the official couple at the end.
George Bernard Shaw was annoyed by fans of his play Pygmalion trying to ship Eliza Doolittle and Professor Higgins, and adamantly declared that the two would never be more than friends (and not very close friends, at that) and declared that, after the events in his play, Eliza Doolittle eventually marries Freddy and they open a flower shop together. He wrote an essay about that, which was included in the 1916 print edition of the play. The musical My Fair Lady, made after Shaw's death, has an ending that implies much more strongly that the two of them do get together. As Pygmalion was written in 1913 and the first film adaptation was made in 1938, this particular Fan-Preferred Couple is Older than Television.
Les MisÚrables: A large portion of fans seem to believe that Eponine is a better match for Marius than Cosette. Admittedly, Cosette doesn't do much in the musical besides hit a B flat and flirt with Marius.
In a lot of respects, that suits Victor Hugo's purpose well: Eponine's role in the original novel was a commentary on the unfairness of class distinctions in contemporary French society, which meant that Eponine/Marius was not 'proper'.
The pairing of Eponine with Enjolras has also been a frequent ship by fans, generally those who prefer not to split Marius and Cosette. Never mind that they never interacted at all in canon.
Enjolras/Grantaire is a very popular pairing despite it being one-sided admiration in the book and stage versions.
Valjean remains a Celibate Hero throughout all versions of the story, but that doesn't stop the numerous fanfics that pair him with FantineorJavert.
Elphaba/Glinda, from Wicked, is somewhat more popular then the other couples. Fiyero/Elphaba has its own large number of shippers though.
One could see Elphaba/Glinda/Fiyero as a kind of fan preferred "triple".
Tracy/Seaweed from Hairspray, rather than Tracy/Link, particularly due to the 2007 film, where Link's role of actually helping Tracy is cut and it's generally felt he did nothing to earn Tracy in the end other than feel sorry about his actions.
There aren't that many Alfred/Sarah shippers out there. Most Tanz Der Vampire fans seem to prefer Sarah with Krolock, and a strong number like to pair Alfred with Herbert. Anyone who's seen the film beforehand ships Krolock/Professor with renewed vigour, some fans going so far as to state that 'Gott ist tot' is aimed at the Professor, not Sarah.
Roughly 90% of what little Company fanfiction there is seems to pair Bobby with Joanne, despite the fact that Joanne is married to Larry. The fact that one of the key scenes in the show has Joanne propositioning Bobby probably helps.
Into the Woods fans love to pair up Cinderella and the Baker. The show basically invites this, since by the end he's been widowed, she's been abandoned, and they're about to move in together with three kids, but nothing in the script suggests more than a platonic relationship.