Zuko's sister Azula gets shipped with everyone she's ever met, as well as many she hasn't. The most common seem to be with Ty Lee, Mai, Zuko, Sokka, Suki and Zhao, but there are many, many more. The idea of Azula having her way with both sides of the Sokka/Suki pairing (sometimes simultaneously) is also fairly common.
Sokka, Fandom Pimp. Forget water or firebending, he's the pheromone bender. He has the benefit of the most in-universe romances—he's Official Couples with Yue and Suki, an infatuation of Ty Lee's, Toph crushes on him to the point that she Accidentally KissesSuki thinking she's him, and arguably has Foe Yay with Azula. He also briefly dated Kaya, a girl from the Earth Kingdom in the canon comic, "Sokka the Avatar."
Fanart made the titular character this before the show even premiered. The fact that she is canonically bi ended up helping a lot.
Iroh II, upon having one line at the very end of an episode, was shipped with every character imaginable in a matter of days.
Opal Beifong quickly joined the list soon after her introduction. Fanart often depicts her with other girls, usually with either Korra, or (aged up) Jinora. But there's still plenty of art pairing her with her canonical boyfriend, Bolin.
Danny from Danny Phantom has been paired with not only almost every single character in the show (Sam, Valerie, Vlad, Dash, Tucker, etc.), but even with inanimate items: Danny/Fenton Thermos anyone? He's paired with characters that don't even exist, and background characters whose names and personalities were invented by fans (such as Wes Weston). He even gets shipped with himself, multiple times over. note Haunted Past (present Danny/evil future Danny) and Pitch Pearl (present Danny's ghost half/present Danny's human half, typically either in an Alternate Universe Fic where they're different people or split via the Fenton Ghost Catcher). The latter even has two named variants - Heroic Amusement, wherin the two halves act the same as they did in the episode "Identity Crisis", and Red Pearl, where Danny's ghost half is being controlled by Freakshow. Crossovers also sometimes occur, such as ones pairing him with Jake Long. Apparently, he's just that compatible.
Raven in Teen Titans, commonly seen shipped with Robin, Beast Boy, Cyborg, Red X, Starfire, Terra, Slade, or others.
There is Craig/Tweek, Craig/Stan, Craig/Kyle, Craig/Kenny, Craig/Cartman, Craig/Clyde, Craig/Wendy, Craig/Mark, Craig/Butters, etc. Lots and lots.
On the female side, there is Bebe, who has been paired with Kyle, Clyde, Wendy, Craig, Kenny, Token, etc. Plus the fandom tends to characterize her as "slutty". Wendy's no slouch in the shipping department, either.
Xandir from Drawn Together. More often than not, he's paired with Captain Hero, but ships with each of the other male characters are not uncommon. In fact, despite him being firmly identified as homosexual, he's often paired in fanfics with the show's female characters.
As of this writing, Webby from Ducktales 2017 has been paired with all three nephews and her new friend Lena.
The Fairly OddParents!, Norm the Genie seems to have started becoming this. Well, he has got paired with a lot of Original Characters, which are either human or genies. An idea that ships him with Wanda has been mentioned on a TV.com messageboard. There is a discussion on FanFiction.Net about whether he's better with Crocker or Imaginary Gary. Also, about him and Princess ManDIE. He has even been paired with Head Pixie.
Both Cosmo and Anti-Cosmo seem to get a lot of this too, even with both of them being happily married in the show.
Riven of Winx Club is the most pair-able character in the whole show.
Bloom has been shipped almost every character in the show both good an evil.
Starscream. All of them. This may be mildly connected to the fact that he's one of the more effeminate characters in a mostly male show in nearly every Transformers series.
Bridgette has been paired up with more than half of the boys at some time or another, though these are less common.
Cody also seems to fit with anyone (though Gwen and Noah are the most common), and his Casanova Wannabe status means that he can believably be paired with any of the female contestants, even the "appearance challenged" ones like Beth, Eva, and Sadie.
Baljeet, but oddly enough, the majority of his are actually canon'; in fact, the only non-canon ship of his is with Isabella. But even then, they have their moments.
Dr. Heinz Doofenshmirtz. The obvious Perry aside, he dated several women on the show. Even ignoring the canon, he has several Fan Girls and Original Characters after him.
Candace, aside from her canon relathionship with Jeremy, with many female characters in the show: Stacy (her best friend), Jenny (her other friend), Vanessa, Isabella (her brother's girlfriend), and Dana.
Mr. Quiet, Mr. Grumpy and Miss Chatterbox from The Mr. Men Show have each been paired with more then a few characters. Miss Calamity's been paired with a lot of characters. Besides Mr. Bump, she's been paired with Mr. Strong, Mr. Happy, Mr. Tickle, Mr. Quiet, and Mr. Scatterbrain.
Jade Chan. Considering she's the only regular female character, she was bound to fill this role.
Not only is Chuckie Finster from Rugrats a big example of this in the fandom, he is also a pretty good example of this in the show. He has had some Ship Tease moments with Lil, Susie and Angelica along with a slightly more than occasional one-shot female character. Tommy Pickles did make an effort to catch up with him in the show (even though it was mainly in All Grown Up!) but Chuckie Finster is still probably one of the more known cases of this in Nickelodean history.
The Flash from the animated version of the Justice League gets this, though granted BatFlash is one of the most popular, he's also been paired with Green Lantern, Wonder Woman, Martian Manhunter, Superman, Supergirl, Harley Quinn, Lex Luthor (though generally not reciprocating any feelings), the Lords!Flash (even though he's, ya know, dead, Fire, Linda Park, Spider-Man...In fact, one of the only ones he doesn't really get paired with is Hawkgirl- probably because canonically they're "Like Brother and Sister".
Double D. Though usually shipped with either Marie, Kevin, or Eddy, he's been shipped with the vast majority of the cast at least once. Yes, even Eddy's Brother.
Nazz has a bit of this too. Gets shipped with Kevin, Edd, Eddy, Sarah, and Marie.
King Julien from The Penguins of Madagascar has been shipped with nearly the entire rest of the cast: Maurice, Mort and all four penguins. On the flip side, he did show interest in the sole leading female, Marlene the otter.
All Hail King Julien adds another fleet with Clover, Karl and Crimson.
According to fanon, this is Arnold's job, as he has a habit of being shipped with EVERYONE. Seriously, EVERYONE. Even Azula. Yay Arnold?
Here is a shipping chart. Note that the four most common pairings all involve Rainbow Dash. And they point to her like an arrow.
Twilight Sparkle also qualifies. In fact it's entirely possible that this trope fits Twilight more than Rainbow Dash! A lot of pics are drawn and a lot of fanfics are written involving Rainbow Dash pairings, but Twilight may possibly have more actual pairings.
Statistically, she only barely made this trope with 1,008 pairings as of April 17, 2013. Rainbow Dash has 1,291. Twilight and Fluttershy tie at 43 for number of unique ships. Dash has a mere 40. 
Equestria Daily has actually graphed the shipping fanfics it's received on two separate occasions. Dash is currently winning on quantity, while Twilight is winning on variety, having been shipped with every one of her friends, herself (twice), her Parental Substitute and mentor Celestia, Big Macintosh, Spike, Luna, Derpy, Trixie, a huge variety of OC's, and on her more "open-minded" days several crossover characters.
There's even a deviantArt group of fans of MordeTwi (Twilight x Mordecai from Regular Show.)
Originally an art commission for a fanfic, Spidertwi has taken off with enough uncommissioned artists to get a Derpibooru tag.
Rainbow Dash shipping is so common that on, Fimfiction.Net, the dedicated MLP fanfiction site, one user, Jake Roberts, tried to coin "Roberts' Razor" which ran: "If a pony exists in MLP, there are at least three stories shipping them with Rainbow Dash." This fell apart when two other users brought up ponies that Rainbow hadn't been shipped with,note For the curious, Fleur de Lis and Celestia but the thought still stands.
As for colts, Big Macintosh seems to be one of the only males the fandom will touch, so he gets shipped with just about everypony at some point... including Applejack... and Apple Bloom!
Soarin is the closest thing to a pegasus equivalent to Big Macintosh (though to a lesser degree), being paired up frequently with Rainbow Dash, Applejack, Spitfire, and miscellaneous background ponies (but not to the same degree of Big Macintosh). Fancy Pants is the most popular Unicorn bachelor, although he's usually only shipped with Rarity and Fleur De Lis.
Caramel is the second most shipped stallion next to Big Macintosh. He's been shipped with everypony in Equestria, and we mean everypony. Particularly impressive since he's a background character.
Braeburn is Big Mac for the slash fans. Everypony is gay for Braeburn. Yes, even the mares. No, this doesn't work.
Because he's the only male main character, Spike is slowly becoming this. He's been paired with each of the Mane 6 at least once, plus the Cutie Mark Crusaders, both Princesses, Little Strongheart, and even Big Macintosh.
Star Wars: The Clone Wars gave us Ahsoka. She's been paired at least with Anakin, Obi-Wan, Padme, Chuchi, Shaak Ti, Yoda, Barriss, Rex, Cad Bane, Boba Fett, Lux, Grievous, anyone.
X-Men: Evolution has Kitty and Rogue. Kitty has her canon pairing with Lance, but also gets shipped with Kurt, Pietro(!), Scott, Piotr, Dani Moonstar, and Rogue. While Rogue has her more common Scott and Gambit pairings, she also is paired with Lance, Pietro, Kitty, and even Wolverine despite the huge age difference in this universe. With these two, it's largely because they're the most popular characters in the show (along with the bellow mentioned Kurt); the female fans either identify with one of them, and the male fans tend to openly crush on them, especially Kitty (Well, the straight male/female fans), meaning they either get paired with whoever the girls personally like, or they get paired with who the boys identify with the most. The same is likely for Kurt bellow.
Interestingly, Jean almost never ends up with anyone other than Scott. Scott/someone else fans generally ignore her or make her Ron the Death Eater. Thought, the occasional Jean/Lance fic and Jean/Logan fics still pop up now and again, so some people like her.
Kurt himself has been shipped with everyone in the cast. Kitty, Wanda, Amanda, Jubilee, Amara, Rahne, Scott, Evan, Logan, Toad, Storm, Lance, and then there's the fact he gets the most fangirls, so there's millions of Original Characters. Chicks really do dig the fuzzy dude (And apparenly the dudes as well). Just say that if you have a name and one line of dialog, someone's going to ship you with Kurt.
This is canon: Dick has been romantically linked with four different characters (Zatanna, Rocket, Batgirl, and Bette Kane) over the series' six-year time frame. Three of whom are his teammates. Throw in the non-canon pairings, like Dick/Wally and Dick/Artemis, and you've got yourself a fandom bicycle.
In a rather odd occurrence, everyone on Metalocalypse tends to be shipped with everyone else, making them all the Fandom Bicycle. Though it doesn't hurt that the Fandom is relatively laid back when it comes to shipping.
Lord Stingray has become second to the Warden in spawning multiple pairings. If he's not simply paired with the Warden as mentioned above, there's Mistress Kilda (his wife that Alice killed), the Mistress (whom he had a failed relationship with), Alice (due to both having slept with the Mistress), Jailbot, and any of the other inmates.
The Mistress herself is a comparatively minor example so far. If not paired with the Warden, Alice, or Stingray, fans will settle for having her paired off with Charise or another member of her staff (less commonly Bruce, rarely Nova).
Dipper of Gravity Falls has been shipped with Wendy, Candy, Mabel, Grunkle Stan, Soos, Gideon, Robbie, Giffany, Pacifica, Tyrone, and even with Bill Cipher, an Eldritch Abomination.
The titular character of The Amazing World of Gumball has a canon love interest in Penny to the point of them becoming an Official Couple but this doesn't stop the Fan Preferred Pairing with him with Carrie. This doesn't stop Gumball from being shipped with Tina Rex, Jamie, Teri, Carmen, Rachel, Molly, Sarah G. Lato, his adopted brother Darwin who fell in love with him when he thought Gumball was a girl, his mother Nicole, and his little sister Anais!
Steven Universe has Pearl, who's already had Ship Tease moments with Garnet and Amethyst and of course her background with Rose, being widely shipped with every Gem introduced thus far (sans Steven) and occasionally some humans.
Later on, Peridot. While Amethyst/Peridot and Lapis/Peridot are most popular, Pearl/Peridot, Garnet/Peridot, Jasper/Peridot are also popular ships. Even Steven/Peridot has its supporters.
Marco Diaz from Star vs. the Forces of Evil gets shipped with seemingly everyone: his best friend Star, his crush Jackie Lynn-Thomas, local bad girl Janna, Star's evil ex-boyfriend Tom, Tad's ex-girlfriend Kelly...
The creators of the show, well known for teasing the fanbase, seem to be both acknowledging and deliberately encouraging this trend by dedicating an entire episode ("Running With Scissors") to Ship Tease between Marco and Hekapoo, a previously-minor character who he'd never even met before.
Lincoln Loud of The Loud House has been shipped with his former classmate Cristina (on which he has or had an unrequited crush), his other classmate Ronnie Anne (actually having feelings for each other helps), the four girls from "Dance Dance Resolution" (all of whom he was set up with on a blind date at the same time), several unnamed female background characters, his best friend Clyde, and, most popular and famously (or infamously depending on one's personal stance), all ten of his sisters. Yes, even his baby sister Lily. That's not even bringing his sisters' male counterparts or his own female counterpart Linka into the equation. Hey, it's in the Theme Song: "One boy and ten girls, wouldn't trade it for the world!"