The 2009 Star Trek gives us Pavel Chekov, aka Ensign Jailbait. Along with Captain Kirk, naturally, but not to the same degree.
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.
The Dark Knight Trilogy: DeviantArt is filled with fanfiction and fanart pairing The Joker with Scarecrow, Commissioner Gordon, Batman, Ramirez, Harley Quinn and the Riddler (neither of whom appeared in the film series), random Mary Sues, Two-Face, and Rachel Dawes.
And that's just the in-universe examples. Outside of that, he's paired with everyone, including Rorschach from Watchmen.
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).
Brought to the nth power after the release of Captain America: The Winter Soldier, for obvious reasons. There, Sam Wilson and Sharon Carter were brought into the fold, while the amount of people who shipped Steve with Natasha went up significantly. Then there's Bucky, where the amount of Fan Fiction and fan art shipping them together skyrocketed. (As of late 2017, Steve/Bucky seems to be the most popular ship in all of MCU fandom, with Steve/Tony not far behind.)
Case in point: AO3 now boasts nearly 100K fics tagged with Steve Rogers (for comparison, Tony appears in 80K fics, Bucky in 60K, Loki 30K.)
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.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...
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 Archive of Our Own fic site even has a tag labeled 'ship Darcy with all the things' that currently brings up 343 stories. Next to Loki, she might be the biggest bicycle in the fandom. 'Darcy is the fandom bicycle and I love it' is nearing 150 fics as well. Darcy even shows up frequently in Crossover Ship pairings, like Darcy/Anders.
Black Widow, stemming from having a lot of this in-universe. Her flirty personality and the fact that she's the most prominent superheroine helps, but she's had some degree of Ship Tease with a large amount of the male cast. In addition to Fan Preferred Couples like Clint and Steve, other popular pairings include the rest of the Avengers, Loki, Bucky, Sam, Pepper, Maria Hill, and of course Darcy. (And after the sequel, Bruce Banner).
With his first appearance in Civil War and the release of his own film, T'Challa/Black Panther has gotten this. Common ships for him include his actual Love Interest Nakia, his enemy Erik Killmonger (which also counts as Kissing Cousins), Steve, Bucky, Sam Wilson, Everett Ross, and M'Baku.
Jupiter and Caine from Jupiter Ascending. If something or someone stands in a room with them, they get shipped with them.
Here's a fun exercise: go to a Ghostbusters (2016) fanfiction site and try to find a story which doesn't ship Jillian Holtzmann with another female character, either from the movie or an original creation. Oh, and remember to take lots of water, otherwise you're likely to perish of dehydration before you find one.
Star Wars features The Heroine Rey, who has subsequently become the source of many ships. Although she is mostly paired with Kylo Ren, it is also common to see her paired with Finn, Poe, Rose,Snoke (despite, or sometimes because of the Squick factor), and more obscure characters such as Jessika, Kaydel Co Connix, and even Captain Phasma. Not to mention her involvement in popular OT3 ships such as Finn/Rey/Poe and Rey/Kylo/Hux.