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. Often with characters named Peter, because that's a Running Gag in the Marvel Universe. And other times with her mentors.
And if there's a popular male character, you can bet someone's shipped them with Northstar.
Storm gets her share as well, on account of attractive character design and copious doses of fetish material, Foe Yay and Les Yay... especially when written by Chris Claremont.
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 Batwoman (lesbian)... and that's about it. Every female who's legal and is remotely in his age range appears to be fair game, whether they're canon love interests (Barbara, Helena, Kory), someone who had a crush on him (Power Girl, Supergirl, Raven, etc), or exist in a different medium altogether. Donna Troy rarely gets shipped with him - fans enjoy their friendship too much - but Donna/Dick has its fans. Even characters that are usually considered to be contemporaries of Bruce and not Dick have been shipped with Dick - examples include Dinah Lance, who once said that Dick was even more handsome than Bruce, and Zatanna Zatara, who was paired with him in Young Justice.
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, like 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.
And then the New 52 went and added Miguel/Tim, Harper/Tim & Cullen/Tim.
When she's not paired with Dick, Barbara Gordon gets a lot of action from other superheroines, mostly from her Birds of Prey team, or Batgirl proteges, and sometimes Dick's own love interests. It doesn't help that Gail Simone intentionally writes in plenty of Les Yay in her runs.
Even Damian, who is theJerk with a Heart of Gold and only ten or thirteen years old (depending on the continuity), has been shipped with nearly everyone in (and out) of his age group and both genders. Between Colin Wilkes, Jon Kent, Irey West, Maya Ducard, Maps Mizoguchi, he has a lot of shipping partners for someone his age.
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, Hercules, 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 paired 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.
The writers seem to be aware of this because since then Loki and Kamala have met in a Crossover that was set at a High-School Dance no less.
As with Mystique above, being a bisexual shapeshifter with both male and female forms who's usually been a villain but also been a hero helps. The logical conclusion to this would be pairing Loki/Mystique...
Looks like new Ghost Rider Robbie Reyes is a good way to becoming this, considering people started pairing him with at least three people (Kamala Khan, Nico Minoru and Water Snake) before the first issue of his series even came out. After his first issue, he got paired with X-23, Agent Venom's sidekick Mania, the Hulk's daughter Lyra, and Jason Todd (who isn't even in the same universe).
Kamala and Loki are often "friendshipped" platonically, perhaps because they're both geeky, web-savvy, and canonically fanfic writers, but actual romance would be squicky since Kamala's only 16 (and Loki's a reincarnation of an ancient god with a very dark past).
Archie himself is obvious. In canon, he practically has his own harem and has had many love interests over the years. In non-canon territory he gets shipped with Jughead and Reggie pretty often.
Jughead is well-known for disliking girls but that doesn't stop him from being shipped with Veronica, Betty, Ethel, and various minor girls. Then there are the people who consider him Ambiguously Gay and ship him with Archie, Reggie, or Kevin.