During the time when Sonic was trying to decide between Mina and Sally, there were flame wars over which was better for him. Things died down a bit when Mina got her own boyfriend and Sonic and Sally got engaged. (Of course, then they went and broke the engagement off...) The fact that Sally was going through her Chickification didn't help, and resulted in numerous hate pages dedicated to what a horrible bitch she was while Mina was clearly the only girl for Sonic.
Batgirl/Nightwing vs. Starfire/Nightwing. That's all. Really not helped by the writers who go Running the Asylum and favoring their own pet pairing while derailing the other. (in example, there's Devin Grayson's canon DFOS against Kory to favor Babs.)
There's still a Terra/Beast Boy vs Raven/Beast Boy war, that was first fueled mainly by Terra 2 and the fans of the cartoon who came to enjoy the comics. Terra 2 was later killed off as cannon fodder in an event, but the wars continued when DC brought out a third version of the character (albeit one not very connected to Beast Boy or the Titans). The second and third Terras have since been erased by the New52 reboot, in which only the first one now exists as an active character. As she's partnered off with Beast Boy, ship wars have started anew. Even more so after Beast Boy winds up separated from that Terra and winds up brought to the Titans by the New 52 version of Raven.
Superman and Lois Lane are a watershed fictional couple in popular culture, so in-and-of-itself, turning back the clock so they were just friends drew some ire, but then pairing him with Wonder Woman it hit a certain Berserk Button for many fans.
Before that, there was a component of fandom who seemed to prefer Lana as Superman's love interest, though it seems to have been localized into Fan Fic. Except that Man of Steel (the comic,not the movie) mastermind John Byrne actually said he wished he could have had Superman and Lana end up together.
Carlie Cooper is not popular with supporters of the Spider-Man/Mary Jane pairing. It seems to go only one way, though, since Carlie doesn't have as established a fanbase yet.
But, for a more even footed warground, try Spidey/MJ vs Spidey/Gwen. Usually, Spidey/Gwen is spared by the others due to her death, but the Spidey/Gwen fans seem to just plain hate Spidey/MJ for happening, since its been cited as the (unconfirmed) reason for Gwen's death. Apparently, that Gwen was becoming boring to the writers and that the suits in charge wanted a major death isn't usually taken into consideration, and some cite OMD as something that needed to happen. However, some MJ/Spidey fans seem to be rather unsympathetic towards the death of Gwen, or will try to rationalize it as a good thing, either because it kickstarted the bronze age, it allowed Marvel to step away from bright eyed idealistic stories, or that Gwen was getting boring.
Don't forget Spidey/MJ Vs. Spidey/Black Cat either.
Now, neither side is helped by the fact Stan Lee has stated he envisioned Gwen Stacy as Spidey's true love, then he goes and has Spidey and MJ get married in his personal comic strip, or how he showed support for Joe Quesada breaking up the couple, only to then put the two back together in his aforementioned comic strip. The guy likes trolling the fans, don't he?
The ship-to-ship combat between Cyclops, Wolverine and Jean Gray is the stuff of legend. Even to this day the combat between Scott/Jean, Logan/Jean, and Scott/Logan can be vicious and brutal, no doubt fueled by X-Films series in the early 2000s. What makes this shipping war particularly unique is that it's actually spilled over in universe, with Logan and Scott frequently trolling one another over their mutual affections for Jean.
And it turns into a four-way battle royale once you include Emma Frost's own relationship with Scott.
Just like her daddy, X-23 gets a fair bit of this, mainly between Helix (Hellion/Laura) fans, Laura/Jubilee fans, and fans who don't see her as having interest in relationships whatsoever. Also toss in Kimura, Gambit, Finesse, Mercury, Dust, Elixir, Fantomex, and more crack pairings such as Wither, Spidey, Hulk, and even Deadpool. There's even more than a few fans who ship her with Daken. And now Rule 63 and time travel is enabling the Scott/Logan Crack Pairing to live vicariously through Laura and Teen Cyke (not that it stops anyone from shipping her with the adult Scott, too). To a lesser extent, the cover preview for ANXM #30 showing Laura and Teen!Warren locked in a passionate in-flight kiss has sparked shipping for that pairing, as well (and more than a few jokes over the idea of Laura moving through the entire O5. Yes, even Jean). In fact it's amazing Quentin Quire's Stepford Cuckoo-induced Imagine Spot of himself flirting with her in Wolverine and the X-Men #4 hasn't created that pairing (yet).