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Launcher Of A Thousand Ships:
: Captain Jack Sparrow. Not at all helped by Jack himself Pirates of the Caribbean flirting with almost everyone.
Jack Frost from is paired with nearly every character in the movie by fans, random Rise of the Guardians Mary Sues, female self-inserts, and then some, so much so that someone made a chart for Jack X (insert character of choice here) ship names. That's not even including the numerous crossover pairings involving Jack.
Elsa from Frozen is shipped with nearly every other character in Frozen, including Hans, Kristoff, her own sister, and characters from other worlds, dimensions, and realities.
Worth noting that her sister Anna shares every single ship. If Elsa is shipped with a character, you can bet Anna has been shipped with the same one.
gives us The 2009 Star Trek film Pavel Chekov, aka Ensign Jailbait. Along with Captain Kirk, naturally, but not to the same degree. (Pavel is becoming something of a Memetic Sex God, if not the inverse of a Memetic Molester. Even Nero's been getting in on the action.)
Gaila, Uhura's Orion roommate, is a fan favorite despite having only a couple of minutes onscreen. In fanfiction and fanart, she's paired with Uhura, Scotty, Sulu, Kirk, and even Spock - especially Spock Prime.
deviantART is filled with fanfiction and fanart pairing Heath Ledger's Joker with Cillian Murphy's Scarecrow, Gary Oldman's Gordon, Christian Bale's Batman, Ramirez, as of yet unintroduced Harley and the Riddler, random Mary Sues, Aaron Eckhart's Two-Face, Maggie Gyllenhaal and Katie Holmes's Rachel, and anything in the Nolanverse that moves. I haven't found a Joker/Alfred one yet, but I wouldn't be shocked. And that's nothing compared to what they do to poor Scarecrow...
And that's just the in-universe examples. Outside of that, he's paired with everyone, including Rorschach from .
Watchmen Kate from is paired with every guy character in fandom, especially since she doesn't hook up with anyone in canon and is perfect for both A Knight's Tale Fan-Preferred Couple and Pair the Spares.
The film version of Sherlock Holmes. He's got Watson, Irene and matches up with Blackwood.
gives us Arthur, who is paired with everyone, the most popular being Arthur/Eames, Arthur/Ariadne and Arthur/Cobb.
Ariadne serves this function for het pairings. The most popular pairings for her are the aforementioned Arthur/Ariadne, Dom/Ariadne, and Ariadne/Fischer.
has The Rocky Horror Picture Show Dr. Frank N. Furter, who has been paired with nearly everyone.
Charlie Dalton from Dead Poets Society.
Probably a combination of being portrayed by Tom Hiddleston and how sad his Start of Darkness is, but Loki has become this from Thor. A brief trip through Fanfiction Dot Net will turn up stories shipping him with Thor; Tony; Steve; Clint; Natasha; Coulson; The Hulk; Sif; Jane; Darcy; Angrboda; Sigyn; Fandral; Baldur; Tyr; Pepper; Amora; Lorelei and even Nick Fury, along with many, many more.
He's even shipped by some with his own actor. That's what happens when you say your character needs a hug...
On a similar note, Tony Stark, who, due to his relationships with all of them in , has been shipped, not only with his The Avengers Love Interest Pepper Potts, but Steve Rogers, Loki, Bruce Banner...
Another one is Steve Rogers, partly because he's played by Chris Evans, and partly because he's either viewed with a lot of "uke" traits (being younger, old-fashioned, and relatively innocent compared to his fellow heroes), or a paragon of manhood who can be strong, protective, and sensitive to other people's imminent emotional breakdowns. His portrayal depends on whoever he's being matched up with (i.e., Peggy Carter, Tony Stark, Bucky Barnes, Loki, Howard Stark, Phil Coulson, etc).
Darcy, with a healthy side order of Ships That Pass In The Night. She's frequently shipped with Loki, Hawkeye, Steve Rogers, Bruce Banner, Bucky Barnes, JARVIS, and Coulson, only one of whom she's even met. On the slash side, she and Natasha are getting shipped together increasingly more often. The Archiveofourown fic site even has a tag labeled 'ship Darcy with all the things' that currently brings up 92 stories. Next to Loki, she might be the biggest bicycle in the fandom. 'Darcy is the fandom bicycle and I love it' is nearing 100 fics as well.