Zuko's sister Azula also 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.
Looking at some fanart, it seems Korra's been shipped with almost every character introduced before the show even premiered.
To an even greater extent, Iroh II, upon having one line at the very end of an episode, was shipped with every character imaginable in a matter of days.
Danny from Danny Phantom paired with not only almost every single cast in the show (Sam, Valerie, Vlad, Dash, etc.), but even with inanimate items: Danny/Fenton Thermos anyone? Apparently, he's just that compatible. He's paired with characters that don't even exist. Danny/Wes, anyone?
He also gets shipped with himself twice over. note Haunted Past and Pitch Pearl.
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.
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.
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.
Also Dr. Heinz Doofenshmirtz. The obvious Perry aside, he dated several women on the show. Even ignoring the canon, he has several Fan Girls and OCs after him.
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.
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.
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.
The fans seem to like shipping within the Mane 6. If one of the Mane 6 is in a shipping story, it's probably with one of the other 5.
Or more than one...
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 O Cs. 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.
The Warden is this in the Superjail! fandom. He has literally been paired with almost EVERYONE or anyone in the series that fans may fancy. The main ships include: Alice (his canonical unrequited crush), the Mistress (his proclaimed arch-enemy that some wish he'd settle down with ), Jared, Lord Stingray, Jailbot (whom, mind you, he sees as a SON figure), some of the inmates (especially Jacknife, Ash note spurred on by the season 3 finale, Gary, etc.) and anyone else fans may think of. Hell, he's even been paired with himself (or at least his future counterpart)! Needless to say, the competition of all the above options can cause much Ship-to-Ship Combat. And then you add in theOnce-ler, fans' own O Cs, or the rare crack pairing with any minor characters other than the ones mentioned before note such as DL Diamond, Charise or any other Ultraprison character, etc., and things get even more complicated to make sense of.
Less frequently, he's paired off with the Twins, though it really doesn't help that people also pair them together or off with their own O Cs.
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).