Back in its heyday, Pirates of the Caribbean fandom was split into approximately three groups: the people who wanted Will/Elizabeth (Willabeth) to happen, the people who wanted Jack/Elizabeth (Sparrabeth) to happen, and the tiny group who completely stayed out of it and just watched the madness, usually complete with pointing and laughing. On one forum, these "debates" got to be so bad that the administrators completely shut down all "shipping debates", and had to relegate the respective shippers to their own discussion threads
And it wasn't just that forum either... fans on a certain website's message board did it too. To extreme levels. There was a point where there was nothing on said forum but shipping "team" threads, and bashing of X character.
Not even Pixar is safe anymore. Ever since Toy Story 3 came out and the amount of Buzz/Jessie shippers increased tenfold, shipping "debates" have sprang up. More extreme Woody/Jessie shippers defend their side till the end, some seemingly convinced that Pixar is intentionally toying with them (no Pun intended). Still, Buzz/Jessie shippers aren't innocent either, as some will jump on any Woody/Jessie piece and decry it non-canon. Then you bring Buzz/Woody into the mix...no wonder people like the idea of a Toy Story OT 3 so much.
Also in the MCU, any mention of the possibility of a love triangle between Thor, Jane, and Sif in Thor 2 has led to snide remarks being traded between Thor x Jane shippers and Thor x Sif shippers, occassionally interrupted by Team Loki. Odin help us all if they actually go with the love triangle in the sequel.
Since its release, Frozen has sparked a rabid shipping fanbase, largely due to Elsa's status as Launcher of a Thousand Ships, the film's limited establishment of canon relationships, the popularity of crossover shipping and the integration of the film into the already chaotic Rise of the Brave Tangled Dragons fandom, and the fact that the film's main cast is fairly small, and thusly, non-crossover ships tend to take on a controversial nature (ask any fan of Elsanna, Helsa or Hanna). As a result, the Frozen fandom is engaged in a brutal series of shipping wars, with the primary combatants being Kristanna (Kristoff/Anna), Jelsa (Jack Frost/Elsa), and Elsanna (Elsa/Anna). Jelsa, Helsa, and Kristanna tend to be more popular with preteens and teenagers while Elsanna is popular with the Periphery Demographic of adults. There's been a bit of ship combat from people who are disgusted at Elsanna, dislike Jack Frost (or just Elsa with males)), can't sympathize with Hans (or can't understand how Elsa could love someone who tried to kill her and her sister), or don't think Kristanna had enough proper development in the film to like.
Probably due to the very small cast, the Maleficent fandom is relatively peaceful - the most common ships are Maleficent/Diaval and Maleficent/Aurora, and they have a different target audience. There is also Aurora/Philipp, which gets along well with Maleficent/Diaval, and, an outlier, Diaval/Philipp, which again has a different target audience than all of the above. The most heated debates are about whether one should ship Maleficent with Aurora - some think it borders on incest as the characters were so close when Aurora was still a child, others say that that's just another excuse to Hide Your Lesbians. As all the shippable (i.e. not evil, non-human or very unimportant) characters are portrayed positively in the movie, there seems to be no debate as to whether people deserve each other - the worst accusation against Philipp is boringness, and in lesbian shipping he's usually allowed to hang around the main characters as best friend. Same with Diaval. And then, of course, there are those who ship all the main characters together..
The first popular ship to emerge from The Force Awakens was Finn/Poe. Fans commented on the bromance between the two, praising their on-screen chemistry. This has brought them at odds with those who think Finn's crush on Rey was cute, and the two groups have gotten into many debates as to whether a same-sex or interracial couple would be more progressive on the big screen. Not long after the film was released, however, Finn's actor, John Boyega, issued a statement declaring Finn/Poe as strictly a bromance. This has not stopped fans from supporting the couple. With Finn/Rey recently sunk via an interview with John Boyega, the potential for a Poe/Finn romance would seem more likely; however, rumours are also circulating that Finn may be receiving a female love-interest in Episode VIII. Only time will tell.
More popularnote that AMV had over a million views at the time of this posting, one month after its creation is the most common Ship Mate to Finn/Poe, Kylo Ren/Rey. The ship became popular after viewers were intrigued by the characters' on-screen tension and sexual chemistry—even Time magazine's review of the film commented on it. However, the ship has gained its share of disapproval, mainly from Finn/Rey fans who don't want to see the heroine paired with the film's Tragic Villain. Some of the nastiest ship-to-ship-combat surrounded Finn/Rey fans calling Kylo/Rey fans racist for supporting that pairing over the interracial Finn/Rey; however, the biggest tool in the anti-shipper's arsenal was that Rey and Kylo had to be Kissing Cousins, based off the theory that Rey is Luke's daughter. When that was virtually sunk via Word of God, they started arguing that a relationship between the two was supporting abuse, which riled shippers into countering that Rey and Kylo are enemies, not an abuser and victim in an established relationship, and there's a fine difference between the two. Even further controversy was sparked when John Boyega (who plays Finn) not only confirmed in an interview that Rey and Finn are just friends, but that the romance in Episode VIII is "not going to go the way you think it's going to go". The anti-shippers have since resigned to arguing that a Rey/Kylo pairing would be too controversial. With over a year left to go until Episode VIII's release, it's safe to say the war has only just begun.
The Star Trek fandom is split between Spock/Uhura, the Official Couple, and Kirk/Spock, the grandaddy of all slash pairings. Some of the fighting has gotten very nasty, resulting in some very homophobic remarks from Spuhura shippers and accusations of Abrams "pandering to middle America" from Spirkers, as well as some racially-tinged Ron the Death Eater directed at Uhura.