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. The fact that this one is usually treated with Fanon Discontinuitydespite fans agreeing with the ship really says a lot.
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.
A large portion of fans seem to believe that Éponine 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: Éponine's role in the original novel was a commentary on the unfairness of class distinctions in contemporary French society, which meant that Éponine/Marius was not "proper".
Éponine/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.
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/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.
In Hamilton, it's fairly common to ship Hamilton's son Philip with Burr's daughter Theodosia, usually on the grounds that it'd be hilarious, considering Burr and Hamilton's relationship.
Death and Elisabeth, the Official Couple of that musical, isn't as popular as Death/Rudolf - the latter has more works on Archive of Our Own and general fan content/discussion than the former. note Okay, Death/Rudolf is canon to an extent, but still... Franz Joseph/Elisabeth, who are canonically married is even worse off, with zero AO3 fics.
A lot of fans of Be More Chill gravitate towards Jeremy/Michael instead of the Official Couple, Jeremy/Christine. While it would be hard to find someone who actually doesn't like Christine, we see Michael and Jeremy hang out far more often, they're already best friends (as opposed to Jeremy just admiring Christine from afar), and Michael himself has been confirmed to be pining for Jeremy throughout the story.