Advice and Trust: Shinji/Asuka/Rei or Rei/Kaworu? Both have been teased multiple times in the story.
The Bridge has started seeing this between supporters of the two most popular potential pairings of the fic — Godzilla/Luna and Godzilla/Blade Dancer, the latter having some evidence and the former only being up to interpretation. Hilariously, the author had no intention of any pairings, but has since started dropping hints just because he finds the whole situation amusing.
Corrin Reacts: Azura/Corrin or Flora/Corrin. The former has the support of the majority of the readers, the latter a significant minority who are aided by the author's hints that he ships the latter too. Of course, there has been the suggestion that Flora, Corrin AND Azura simply all get together.
The Great Meta Fic was based on a contest called "Ship Survivor" on the RWBY subreddit, and has ships from RWBY as the basis of nations.
Hivefled: Nepeta dated and broke up with Karkat before the main story happened, and is horrified at Lereal's suggestion they follow in their ancestors' footsteps by dumping all their other quadrants for each other. Karkat's not best pleased either.
Living in The Vampire Diaries has wars over which OC Damon should be with. Some want him with Tori, where the others want him with Sidney (even they're supposed to be Just Friends). The fact Tori is with Stefan while Sidney is single doesn't help, nor does Damon being Sidney's sire
Paradise Thwarted: This happens In-Universe with the denizens of the Fangirl Layer of Hell. Factions dedicated to different ships spend all their times fighting wars with other factions over whose ship is better.
When Ash's Johto and Unova team meet, they immediately start arguing about whether Misty or Iris was a better fit for Ash, advantages ranging from Iris being more available to travel where Misty is 'confined' to the gym to Misty being more comfortable with her available titles while Iris has vaguer goals. The only two not shown to take part in the 'debate' are Sandile and Heracross, Sandile aware that Ash doesn't want to get involved with any of the girls and Heracross forcing the others to focus on the immediate priority of getting back to Ash.
While Ash is in Orre, Serena starts to suggest to the others that they need to take steps to avoid this, as Ash's fragile mental state means that all of the girls will be needed to keep him from killing himself, and they can't risk Ash being provoked by finding any of them arguing.
Project JPDE, as it is currently in the development stage with several new route events being made every month, has had a number of these with regards to who will get nominated that month. In particular, the March 2019 public nominations caused a massive explosion between fans of Nora Valkyrie and fans of Titania, resulting in both groups campaigning all across social media in an attempt to drum up support for their side. (Nora ended up winning by 1 vote.)
Red, an Arrow fanfic, has Thea/Roy shippers vs Thea/Sara. One is based in canon with the other having no indication that they even know each other, but has a more BirdsOf a Feather vibe with both being vigilantes.
The Unity Saga: This happens Seven of Nine kills Mara Jade, after both parties are established as loving Luke.
In "I Am Not a Victim", the first fic in the Valkyrie On Fire series, Cinna essentially weaponises this when he anonymously creates a mass of merchandise focused on the idea of Katniss with Peeta and/or Glimmer during the Games, encouraging the Capitol citizens to root for Katniss to end up with one or the other. This actually culminates in a near-riot when the Games organisers dont immediately announce that the contest is over or that the new rule is invalid after Peetas death as the fans resent the implication that neither of their ships were ever going to be allowed to win.
Zelda and the Manacle of Cahla: Done In-Universe. When Zelda first arrives in Zora's Domain, the royal priestess sisters, Asine and Asuca, are in fierce completion over who will marry Prince Tyrus. Most of the tribe has split into two factions, one pro Asine/Tyrus and the other pro Asuca/Tyrus, and is on the brink of physical violence. Zelda helps Asine and Asuca admit the fighting was more about one upping the other than love for Tyrus, after which they reconcile and call off all engagements. The factions calm down and shrink in members, while some Zoras become pro Zelda/Tyrus.