In the context of fandom, "shipping" is when fans support a specific coupling in a TV show, movie or comic book ("shipping" being short for "relationship"). It doesn't technically have to be a romantic relationship (you can ship friendships) but for the most part, we're talking romantic ones here. Shipping is extremely common for fans of any work of fiction and superhero fiction is no different. The trick, though, is what happens when there are multiple possible relationships for the main character to pursue.
In that scenario, you get ship wars, as various sects of the fandom clash over which is the "best" ship possible. Note that the fans involved in these ships generally get that the ship that they're hoping for won't ever actually occur (Steve Rogers and Tony Stark, for instance, are not actually going to become a couple), but that doesn't mean that it isn't still fun to support the idea. Since we're only talking ships that have people taking opposite sides of the debate, we're not counting some popular ships that really have no notable rivals, like Deadpool/Spider-Man, Charles Xavier/Erik Lensherr and Maggie Sawyer/Alex Danvers. Actually, Sara Lance/Alex Danvers is gaining some steam as of late, but it's nowhere near "Sanvers." Here are some other popular ships that almost cause their own civil wars.
15 KITTY PRYDE/COLOSSUS VS. KITTY PRYDE/STAR-LORD AND COLOSSUS/DOMINO
Kitty Pryde and Colossus are one of the odder "one true pairings" in comics, considering they began dating when Kitty Pryde was barely a teenager and Colossus was a young adult. However, after breaking up following Secret Wars, they got back together following Colossus' return to life in Joss Whedon and John Cassaday's Astonishing X-Men run. They broke up again after Colossus briefly became the new Juggernaut.
After their break-up, Kitty entered into a serious relationship with Peter Quill, Star-Lord. Kitty even joined the Guardians of the Galaxy and got engaged to Quill! Meanwhile, after being one of the X-Men who merged with the Phoenix Force, Colossus was a mess and ended up dating Domino. Kitty and Colossus are both X-Men together again and fans wait for them to eventually get back together. Colossus/Domino fans, though, are surprisingly resilient, considering how relatively short their relationship was.
14 JOHN STEWART/HAWKGIRL VS. JOHN STEWART/VIXEN
When the Justice League cartoon began, most of the League were very well known to the public and their "shipping" was logically kept to a minimum. The notable exceptions to this were Green Lantern (John Stewart) and Hawkgirl, whose backgrounds were relatively free to be explored by the writers. Thus, John and Hawkgirl slowly but surely developed a relationship.
Hawkgirl left the League, however, after she helped take part of a Thanagarian invasion of Earth (she ultimately sided with the League against her own people). While she was away, John began dating new League member, Vixen. When Hawkgirl returned to the team, she and Vixen seemed poised to fight over John, but instead they just shared a drink together. "Shipping" was less of a big deal when the Justice League was on the air or else these two ships would be much bigger deals.
13 SUPERMAN/WONDER WOMAN VS. SUPERMAN/LOIS LANE
Surprisingly enough, Superman/Wonder Woman was rarely a relationship that was explored in the comics. In fact, Wonder Woman and Superman rarely even teamed up with each other for their first 30 years as characters. It was not until Wonder Woman lost her powers in the early 1970s that she was first presented as a possible love interest for Superman.
In the New 52, however, with Superman and Lois Lane's marriage wiped out of continuity, Superman and Wonder Woman were finally free to date in the comics. They became a couple and even shared an ongoing series together. However, Superman ultimately ended up back with Lois Lane, as the Pre-New 52 Superman and Lois (still married) were introduced and that version of Superman became the "main" Superman in the DC Universe.
12 BARRY/IRIS VS. BARRY/KARA, BARRY/KAITLIN AND BARRY/FELICITY
In the world of the Flash, there really was never any other relationship pursued by the show's creators than Barry Allen and Iris West. Early on, we got to see an article in the future that was written by Iris West-Allen, so it was clear that they were the "one true pairing" of the series. However, a problem arose, which was that Grant Gustin had crazy good chemistry with everyone.
So while Barry/Iris was always going to be the final deal, there was fan support for Barry to be shipped with Supergirl, Caitlin Snow and Felicity Smoak. The Barry/Supergirl shippers got an extra dose of fun when Flash and Supergirl crossed over for a musical episode, as both Gustin and Melissa Benoist (Supergirl) were great singers (they were both even on Glee together) and they sang together a lot.
11 SUPERMAN/LOIS LANE VS. SUPERMAN/LANA LANG
The original superhero "ship war" occurred in the pages of Superman's spinoff title, Superman's Girl Friend, Lois Lane, and it starred Lois Lane and Lana Lang, who had been introduced years earlier as Superman's love interest when Superman was a teenager (and going by the name Superboy). Now an adult, Lana Lang traveled to Metropolis to also find work as a reporter and she and Lois constantly fought over Superman in the pages of Lois' spinoff comic book.
Years later, when Warner Bros. did a new series set during Superman's teen years, called Smallville, Lana was again his main love interest. However, fans were also drawn to the friendship between Clark Kent and his fellow high school journalist, Chloe Sullivan. Then, of course, the show introduced Lois Lane and she slowly but surely took over as the main love interest on the series.
10 NIGHTWING/STARFIRE VS. NIGHTWING/BARBARA GORDON
It wasn't until the early 1980s, when Robin met the alien princess, Starfire, in New Teen Titans, that Dick Grayson had a longterm girlfriend. Their romance charted new territories for superhero romance (in other words, we actually got to see them share a bed together). They seemed like they were going to be together for the long haul, with Dick even changing his name to Nightwing to get away from the Batman titles.
Then the Batman Animated Series came around. Batgirl had never been a love interest for Robin until that cartoon series and soon, the comics retroactively made them love interests and now Nightwing (back with the Batman titles and broken up from Starfire) pursued a relationship with Barbara Gordon, the former Batgirl who was now Oracle. There were many bumps in the road along the way (like the revelation that Dick slept with Barbara while engaged to Starfire).
9 HARLEY QUINN/POISON IVY VS. HARLEY QUINN/JOKER
Harley Quinn was literally designed to be part of Joker's supporting cast. However, once she was introduced as a part of the Batman Animated Series, fans soon fell in love with her and the show's creative team began developing the character further. When she was then paired with Poison Ivy in an episode, there was an instant spark and soon the show explored the Harley/Ivy relationship a lot over the years.
That carried over into the comics (as Harley was introduced into the main DC Universe), but Harley was always true to "Mistah J." However, in recent years, her feelings for Ivy have become stronger and they have become one of the most shipped comic book couples on the internet (while their relationship has become more explicitly romantic in the comics to go along with the increased fandom).
8 BATMAN/SUPERMAN VS. BATMAN/CATWOMAN AND SUPERMAN/LOIS LANE
As the most popular superhero friendship (having been a significant part of their titles since the 1950s), it is only natural that Batman and Superman have also developed into one of the most significant ships on the internet (they are probably second only to Spider-Man/Deadpool among comic book-only superhero ships). Films like Batman v Superman: Dawn of Justice only ramped things up, as their tension could easily be construed as sexual in nature by some fans.
However, there still remains a large group of fans who ship Batman and Superman with their "traditional" partners, Catwoman and Lois Lane, respectively. So it is not like there aren't significant alternate ships out there for the heroes. Of the two, Batman and Catwoman get a little more support in the fandom than Superman/Lois Lane.
7 STEVE ROGERS/BUCKY VS. STEVE ROGERS/TONY STARK
As soon as the first Captain America film came out, fans were drawn to the close friendship between Steve Rogers and his childhood friend, Bucky Barnes. That friendship became the centerpiece of the next Captain America film, Captain America: The Winter Soldier, as Cap risked it all to save his old friend from his new life as a brainwashed Russian assassin.
Then Captain America: Civil War came out, and suddenly it was practically a big ol' superhero romance played out on the screen, only with ostensibly straight male leads. Once again, Cap must risk everything to save Bucky (who has been falsely accused of assassinating the King of Wakanda) and Steve is forced into being at odds against his close friend, Tony Stark. Tony's classic (almost tearful) response to Steve's "I had no other choice. He's my friend" defense ("So was I," replied Stark, hurt) was the stuff that fandoms were built upon.
6 KARA ZOR-EL/MON-EL VS. KARA ZOR-EL/LENA LUTHOR
Besides a surprising amount of chemistry with the Flash, Supergirl's first season did not have much in the way of fandom-supported relationships. Sure, there was the prominent "Will they or won't they?" between Kara Zor-El and James Olsen, but that relationship didn't inspire a whole lot of shipping on the internet.
In the second season, though, Kara met Mon-El and they became a prominent couple on the show and in fandom (Karamel). We also met Lex Luthor's sister, Lena Luthor, who developed a close friendship with Supergirl throughout the season, as Lena struggled to do some good with the Luthor legacy. Fans, meanwhile, lost their collective minds. The "SuperCorp" ship between Kara and Lena became probably the most popular superhero ship of 2017. The show courted controversy when Supergirl cast member Jeremy Jordan mocked the intensity of "SuperCorp" fans at a cast interview. Fandom was not pleased.
5 CYCLOPS/JEAN GREY VS. CYCLOPS/EMMA FROST
When Grant Morrison took over as the head "architect" of the X-Men titles in 2001, he introduced a number of major shake-ups. He had the mutant population explode, he had the mutant nation of Genosha destroyed, he had Charles Xavier publicly "out" himself as a mutant and, perhaps most notably (in the eyes of some fans), he ultimately broke up Cyclops and Jean Grey.
Morrison felt that Cyclops and Jean seemed more like best friends than actually all that attracted to each other and he brought in Emma Frost as a romantic foil for the couple. When Jean was then killed at the end of Morrison's run, he had Jean actually push Cyclops and Emma Frost to get together after her death. Despite her endorsement, many fans were not having the new coupling and a "ship war" has remained in effect for over a decade since.
4 TONY STARK/BRUCE BANNER VS. TONY STARK/STEVE ROGERS
As noted, Tony Stark and Steve Rogers inspired a great deal of fandom support, under the ship name of "Stony." That support, though, ultimately paled in comparison to the support that Tony Stark drew from fandom for the ship between Tony and Bruce Banner ("Brony"). In fact, the whole love triangle in fandom has been dubbed "Stark Spangled Banner."
The shipping for Tony and Banner has varied from mere friendship (the depiction of them as "Science Bros" doing cool science together) to far more romantic pairings, as well. In a famous talk show appearance, Robert Downey Jr. and Mark Ruffalo gave the fans a thrill by acting out one of their shipping poses as the actor who plays Steve Rogers, Chris Evans, stood behind them the whole time.
3 SPIDER-MAN/GWEN STACY VS. SPIDER-MAN/MARY JANE
One of the oddest ship wars in superhero comics did not really take place until one third of the rivalry was actually dead in the comics! The rivalry between Gwen Stacy and Mary Jane over Peter Parker was remarkably one-sided when all three of them were alive, as Peter treated Gwen as his "one true love" in the comic and Mary Jane was almost a distraction.
Gerry Conway, though, preferred Mary Jane, so he helped to write Gwen off by killing her in a classic early 1970s storyline. When Conway left the series, though, Mary Jane was eventually written off, as well. She was brought back in the 1980s and then she and Peter married. Since that point, there has been a number of debates as to which pairing should be the one to make it into TV shows and movies based on the comic.
2 CYCLOPS/JEAN GREY VS. WOLVERINE/JEAN GREY
Like Gwen/Mary Jane/Peter, the love triangle between Cyclops, Wolverine and Jean Grey was decidedly uninteresting for the first decade of its existence. Jean loved Cyclops, Cyclops loved Jean and Wolverine was just the sort of creepy stalker who loved Jean from afar. When Jean was brought back to life in the mid-1980s, however, Chris Claremont began to retroactively reveal (through stories set in the past) that Jean was actually really attracted to Wolverine (despite showing no signs back in the original comic books).
Still, Jean and Cyclops' relationship continued on (only briefly interrupted by her death and Cyclops marrying another woman) and they got married (Wolverine did not attend the wedding). However, that did not stop the attraction from continuing to grow and now that Jean Grey is set to return, who knows if she will get together with Wolverine or not.
1 OLIVER QUEEN/FELICITY SMOAK VS. OLIVER QUEEN/LAUREL LANCE
As you might have noticed, most popular ships in fandom are those that have no real chance of actually occurring. Therefore, when a ship actually does have a chance, fandom really goes nuts over it and that was the case with Olicity. When Arrow debuted, it was clear that the "one true pairing" on the show was Oliver Queen and his ex-girlfriend, Laurel Lance (who was destined to become Black Canary, who is Green Arrow's "one true pairing" in the comics).
Then a guest star in an early episode stunned everyone with her chemistry with Oliver and suddenly Felicity Smoak was a regular cast member and "Olicity" became the most popular ship in fandom. Then the backlash occurred when fans felt "Olicity" had taken over the show and when Laurel was actually killed off, fans went ballistic. In the end, though, Oliver and Felicity ended up getting married.