For a long time (and probably to this day) Gambit has fulfilled this role in X-Men fandom, being paired with any and every character out there, regardless of whether he likes them, or whether he's even met them in the first place.
Mystique, a bisexual shapeshifter who's taken on both male and female forms. She's usually a villain, but has spent time as a hero, so she can be paired with anyone of any Character Alignment, gender, or sexual orientation.
Kitty Pryde as well.
And if they there's a popular male character, you can bet someone's shipped them with Northstar.
It's a running gag in the DC Universe that virtually every female that comes in contact with Nightwing falls in love with him. And that's not even getting into the Foe Yay...
Known exceptions: Dr. Leslie Tompkins (Old enough to be his grandmother) and Stephanie Brown (underage, and dating his little brother).
Dick's "bicycliness" doesn't stop with the females he meets, at least in the fandom. Popular slash pairings fix him up with Bruce, Roy, Wally, Tim, Jason, Terry...
Bruce. Look at how many love interests he's had in the course of Batman History! Plus how many women have kissed him! Even non-love interests have kissed him! (Harley Quinn) If you think about it, comic book love interests, villainesses, movie love interests, TV show love interests, animated DCU love interests... that's just the TIP of the iceberg! Plus slash pairings! IN ALL INCARNATIONS!
Tim Drake! He stays mostly to the batfamily side of things, but popular ships include: Dick/Tim, Jason/Tim, Bruce/Tim, Damian/Tim, Kon/Tim, Bart/Tim, Kon/Bart/Tim, Steph/Tim, Tim/Tam, Cassie/Tim, Cass/Tim, Ra's/Tim and Tim/Bernard. Canonically, he is a gorgeous woman as Caroline Hill (one of his many undercover disguises), and there are several fics pairing other characters with "her." It also helps that Marcus To's drawings of Tim brought him into the ranks of a Bishōnen with pretty hair.
There's even one fanfic out there pairing him with HalEmmerich!
Clint Barton is largely considered the Marvel character who can be with any single other superheroine in the Marvel Universe and it not to come off strange. His current tally includes the first Ronin (Maya Lopez), The Black Widow, Mockingbird, Moonstone, the Scarlet Witch, and the Wasp. Currently in an "it's complicated" relationship with Spider-Woman.
Wolverine is considered to be a sexual stud-muffin given that he has been paired with virtually every other character possible in the X-men in fanfic. The authors sense this too. His canonical love interests have included: Silver Fox, Viper (with a very descriptive honeymoon), Jean Grey (never consumated), Mystique, Storm, the Heather Hudson version of Vindicator, the Silver Samurai's sister Mariko Yashida, his string of female sidekicks like Kitty Pryde, Jubilee, and Pixie, and many more.
The Black Widow is the female version of this in Marvel comics. Her relationships include Iron Man, Hawkeye, Daredevil, Bucky Barnes, an attempt by the writers to set her up with Deadpool(?!), and arguably Nick Fury. Even her late husband (she's a widow, get it?), the Red Guardian, was a superhero. Yes, Natasha goes with everything.
Called out in one of the early issues of her recent solo series, where Captain America, Iron Man, and at least one other are protesting her innocence to a higher-up, who concludes his devastating rebuttal about her super-spy skills with "Case in point. . . is there any one of you she hasn't slept with?" Cue three very guilty looks.
Wonder Woman is also considered to be a Launcher of a Thousand Ships for the main heroes of DC comics. It's just that none of the pairings ever stick. They've tried to pair her up with Aquaman, Batman, Superman, and Nemesis.
Also, as of The New 52's Suicide Squad, is frequently ship teased with Deadshot.
Daken, Wolverine's son and member of the Dark Avengers, is starting to become this, mostly just being paried with male characters though. But then again he's a Depraved Bisexual whose powers include super-healing and what amounts to sex-pollen. Common pairings include every member of the Dark Avengers (Bullseye and Mac/Venom in particular), his father Wolverine, Northstar, Deadpool, Anole, Wiccan and Hulkling.
While not getting much love in-universe, Deadpool fans seem to think he has chemistry with almost anyone who tolerates him for more than 5 minutes. Popular ships include Cable, Weasel, Bob, Domino, Black Widow, Spider-Man, Bullseye, and Captain America.
Back in New X-Men Prodigy had only been shipped with his canonical Love Interest, Surge, and possibly with Wallflower. Since he joined Young Avengers however, he has also been paired up with Speed, Hulkling, Miss America, Kate Bishop, Loki and Transonic.
Since Plot-Relevant Age-Up Loki himself gets shipped with pretty much every single member of the team.
Looks like Robbie Reyes, new Ghost Rider is on a good way to become this, considering people started pairing him with at least two people (Kamala Khan and Nico Minoru) months before first issue of his series even came out.