The eponymous character and her canon love interest are paired with nearly everyone and a huge group even in canon. Depending on the version, Usagi is somehow connected romantically with Mamoru, Ali, Demand, Haruka, possibly Rei, Manga!Galaxia and Seiya. Mamoru has Rei, Beryl, Hina, En, Black Lady. That's not even counting the crushes and fandom pairings.
Being the daughter of said couple, Chibi-Usa has quite a few love interests as well. Helios and Perle are canon, Kyuusuke is implied to have a crush on her before he got together with her best friend Momoko (similar to Umino and Naru in relation to Usagi), and there are quite a few who think she has something more than friendship with Hotaru.
Motoki would also count in his anime and manga incarnations as Usagi, Makoto, and Minako all have crushes on him in both canons, while Ami has a crush on him in the manga and Rei has a crush on him in the anime. Plus his fiance, Reika.
Kagome, who gets crossed over and paired up with so many characters from so many different fandoms that it would be impossible to list them all (usually the character the author wants to hook up with), due to her being the character easiest for a teenage girl to relate to. Then again, she can travel through time, so one's Willing Suspension of Disbelief can be stretched just a wee bit here.
Ditto for Sesshomaru, due to him being the most lusted-after male character, girls tend to pair him with whichever female character reminds them of themselves (usually Kagome).
Ranma ½: Ranma has this problem and is shipped with EVERYBODY. And not even that, when he's a girl with a male heart you can expect the obvious. Shipping Ranma with everybody isn't even restricted to his own universe! Ranma ˝ is crossovered a lot in fanfiction. That means more characters to use which in turn means more characters to ship Ranma with, regardless of gender. He's the ultimate fandom bicycle. Everyone and we do mean everyone gets a ride.
Mobile Suit Gundam Wing: While most popular pairings happen to be Heero/Duo (and to a lesser extent Heero/Relena) and Trowa/Quatre, you won't have much trouble finding a rather large amount of any combination of pairings amongst the pilots, with another smaller subset involving the other characters. There is a good chunk of stories about all five of them that aren't crackfics. Duo Maxwell is often decpited as "uke" or a playboy in fanfiction and in addition, or perhaps subsequently, happens to be on the receiving end of this trope fairly often. You will see Heero often paired off with characters outside the Gundam Wing universe.
Mai-HiME and Mai-Otome: Shizuru leads the way in a fandom filled to the brim with yuri fanfic and fan art, having been paired up with the likes of Natsuki, Nao, Mai, Haruka, Arika, and Tomoe.
Sakura and Hinata appear to have this ability to be shipped with anyone. Besides the usual Sasuke/Sakura and Naruto/Hinata pairings, you also have stuff like Kakashi/Sakura, Itachi/Sakura, Sai/Sakura, Neji/Hinata, Sasuke/Hinata, Shikamaru/Hinata...and they have even been shipped together.'
On the male side, in non-yaoi pairings, there are many fanfics and Doujin putting Shikamaru with Ino, Sakura, Hinata, Temari, Shiho, his teachers's lover along with many others. Sometimes, all at once. There probably isn't a single person in the entire show with whom this guy hasn't been paired.
Naruto himself has been shipped with nearly every named female in the series (and a lot of the males, too), sometimes all at once. Like ships that include the Nine-Tailed Fox! And then there's FemNaruto who ends up paired with most of the male AND female cast. Or gets paired with people from other series.
Kakashi seems to get paired with everyone in the show too.
Negi. Although the majority are justified as roughly 90% of females he meets develop crushes on him. And when we say 90%, we're talking about a series that introduced over 30 characters in chapter one. At this point, he's kissed half of them. Somewhat lampshaded by Chamo, who appears to see Negi as a Launcher of a Thousand Ships. He's somewhat biased, as he has financial incentive to make Negi go for as many as possible; he earns several thousand Ermine Dollars every time Negi kisses a new girl.
Invoked by Paru describing Yue's situation, gaining a respectable female harem. Yue is not amused by this commentary.
Tsuna. There was Spanner/Tsuna doujinshi on sale merely weeks after Spanner's introduction. Japan moves in mysterious ways. And Tsuna is joked to be the fandom bicycle as well. Funnily enough, it's actually rare to see fan merchandise of Tsuna with any of the canon love interests (the four girls, to be precise). The magical Tsuna bicycle starts getting more outrageous when one realizes that there are fandoms that support Squalo x Tsuna and Bel x Tsuna.
Gokudera Hayato gets paired with not only every guardian, but with the TYL!Guardians as well (oh parallel world sex), and with multiple enemies AND paired with the girls. (Including his sister. Disturbing yes?)
Chrome's paired with freakin' everyone: Mukuro for obvious reasons, Tsuna cus she kissed him when they first met (that still hasn't been explained), Hibari cus he saved her life once or twice, Ken cus he's Tsundere towards her, Chikusa cus he's on the same team as her, MM cus of the Foe Yay, Mammon cus of his Naughty Tentacles, and Julie on account of the kidnapping, etc etc. These all make some sense and at least have had vague screentime. It starts getting weird when it starts being characters she's never talked to or even talked about. Prime examples include Enma, Genkishi, Bel and for some reason Levi...
Haru, who even has a fanclub on deviantArt specifically titled the "Anyonex Haru-Club"
Ash has been paired with everything from every female lead, to every male lead, to Pikachu, to Bayleef, to Latias (who if you want to get technical, is actually tied for the furthest anyone's ever gotten with him.) Have some actual numbers: As of October 2011, the list referenced below contains 495 named ships involving Ash. Plus 11 involving Ashley (Ash as a Wholesome Crossdresser). Plus 13 involving Ashachu (Ash as a Pikachu (it's canon)).
Name any two characters — Pokémon included — in the anime, and not only will there be a shipping fanbase, it will have a name! And it's not just limited to living characters! There is one called DownBoyShipping that consists of Misty's hand and Brock's ear!
Ash's mother Delia also gets this treatment. (Just look at all of the "Who is Ash's Father" theories in WMG) The two most popular choices are Professor Oak and Giovanni, but there are lots of other possible candidates.
Meiling and Touya of Cardcaptor Sakura, while mild examples compared to other fandoms, get treated this way in fanfiction; especially Touya, who is most often targeted for the fandom's few common crack pairings.
Zoro himself is beginning to get a harem of his own. Not only is he commonly shipped with Nami, Robin, Sanji, Luffy, Ace, Mihawk, and Tashigi, but you can also add in now Bonney and Perona!
Shanks gets shipped around a bit too, just not as much as the other listed characters.
Alviss from Märchen Awakens Romance. The fandom isn't huge, but the boy has been paired not only with every kind of Mary Sue imaginable but with his teammates and the villains from the show. Especially because every single pairing has been hinted in canon. Even the evil overlord has attempted to kiss him in the anime, and it was a kids' show, with not even a drop of blood.
You have five seconds after finishing this sentence to name a female character from Hayate the Combat Butler with whom Hayate isn't paired.......That's what I thought. Doesn't hurt that most of them are canon◊. And the guys aren't always exempt from that.
France: He gets some with everyone, and some fanworks just list France/Everybody as one of the included pairings.
In Japanese fandom, Japan gets paired with everyone. 'Everyone' including the "well, that does make a fair amount of sense" ones like England, America, Greece (which might actually be the most canon one on this list), Italy, China, and the "where did THAT come from?!" ones like Switzerland, Norway, etc. He's not nearly as much of one in the English fandom, where England and Greece are the only ones he's paired up with any regularity, but he has definitely gotten fics with America and [insert any other Asian nation here]. For an idea of how much this trope applies to him, this Japanese fanart◊ shows only the more plausible ships with him. It still takes a while to get through all of them.
The English-speaking fandom does the same with America to a slightly lesser extent. 'Slightly lesser' meaning that there's more of a tendency to pair him with select characters that have the most canon/historical basis with him (England, Russia, and Canada) than with apparently random characters. Keyword being slightly: you still see him being paired every now and then with the likes of France, Japan, Prussia, Vietnam, etc. He's not as much of one in the Japanese fandom, naturally, but considering that there are tags on Pixiv for pairings like America/Spain, America/Taiwan, and America/Seychelles...
Similarly, Vivio enjoys love from all sides as well. It is particularly ironic that, as the daughter of the greatest shipwar trigger in the show, so far there have been no shipwars regarding Vivio. Though this may partially be because Vivio lacks any pairing fanservice, and thus is still fair game for even the most hardy of shippers.
With Vivid Memories ongoing, shippers are mostly aiming for the girls in Vivio's age group. Two of which are the same as her: reincarnates of ancient Belkan Lords.
The only qualification for you to be shipped with Hayate is that you must be female and you must exist in-universe. The former depends on how much you want to stick with the Les Yay theme of the show (though even if you do, Zafira is still one of her knights), and the latter depends on your opinion on Cross Overs.
In Code Geass, Lelouch is paired up with everybody, including his siblings, best friends, enemies, acquaintances, and people he's probably never even met. And sometimes with more than one at a time. Occasionally one comes across selfcest for poor Lelouch, where he's paired up with Zero.
Rock is also guilty here. There are fics of him getting it on for at more than one night with Revy, Eda (both at once at one point), Shenhua, Balalaika, and Roberta. Who knew a scrawny former office drone could be such a badassand a chick magnet?
Hikari as well; she's been shipped with Daisuke, Takeru, Taichi, Yamato, Ken, Koushiro, and characters from other series. In a Digimon crossover, whenever there's a major male character in the other series who will be shipped with a Digimon character, said latter character is probably going to be Hikari.
In Digimon Tamers, there is Rika Nonaka/Ruki Makino. Every single possibility has a fanbase. Rika/Takato, hotblooded girl and sweet boy who can help each other soften/toughen up, respectively (this is the one with the most character derailment, as Rika often becomes a backboneless Distressed Damsel for Takato to save in fanfic. Most popular evidence: Takato's dream of her in the premiere.) Rika/Henry (Jenrya, Jianliang, et cetera), hotblooded girl and calm, levelheaded boy (most popular evidence: any of the brief moments where she'd try to properly butt heads with him and it didn't work. The balance would do them - mostly her - some good, it's held.) Rika/Jeri (Juri), not just 'cause Girl on Girl Is Hot: Jeri makes a good damsel in distress for Rika, and could be someone she'd take under her wing (the one softens/one toughens dymanic is present in almost every ship for her.) Rika and Renamon - between Renamon's sultry voice and Cute Monster Girl-ness, Girl on Girl Is Hotdefinitely applies with the two even if Renamon's not humanoid. There's also Renamon's total devotion to Rika, and it clearly going both ways (Rika not being the type to constantly state it, but it's just as obviously there.) Now, as a Digimon and human partner this is expected, but Renamon is by far the most fleshed-out Digimon partner character, so it shows with her that it's every bit as voluntary as it would be with two humans, and not just "the critter follows the human around 'cause that's how an A Boy and His X series goes." (It's considered to be true with all partners in all series, but with Renamon it shows in every episode.) Finally, Ryo Akiyama, only person ever to beat her at the card game, and with a great love of flirtatious banter usually directed at her, with her being annoyed by it all - the best in Slap-Slap-Kiss, every scene in which the two are together are considered evidence; unlike so many of the others, there's no one scene you can remove and many an Official Couple starts out the way they started; Digimon was just never as romance-oriented a series as fandom wants it to be. About the only pairings for her without a fanbase are her and Kazu or Kenta (Those Two Guys, paired with each other or no one.)
Henry himself is shipped with Rika (see above), Takato (they've got one really good Ho Yay moment, in addition to the two male leads being the easiest choice for Slash Fic in any series), and Renamon (they have two really good Interspecies Romance-yay moments.) Three seems small, but this cast is smaller than Digimon Adventure. Not shipped with him are only Jeri (Takato's canon love interest who has limited interaction with Henry), Ryo (Rika's arguably-canon love interest who has limited interaction with Henry), Those Two Guys (see above), and the Mons with childlike personalities, or physical characteristics that would make them harder for most to see as a 'ship (maybe he'd get along with deliberately-ZeerustTin Can Robot Gardromon, but the question isn't often asked.)
Everyone wants to bang Light, and to be honest there is evidence for many of them (just check out the Death Note Ho Yay page). On the het side there is Misa and Kiyomi who are both in canon love him. There's also Naomi who he charms into giving her real name. On the slash side there is the largest pairing of all, Light and L, as well as Mikiami, Near, Matsuda and even Mello and Matt who never met him in person but were following him. Then there is the squicky ones like Sayu and most strangely popular of all, Ryuk.
L. While the most popular is of course him and Light there is popular fan theorising that Near's idolising of him is a diluted version of BB's obsession with L, which is also a fairly popular pairing. He's also shipped with Aiber (who he has a history with) and Matsuda (who fanboys over both L and Light). For the few Het Shippers in Death Note there is Naomi who has discussed in canon if she fancies L, Misa ("I could actually fall for you"), and Wedy. Then there are the cracktastic stories of LxCake, though that too may be canon. And these are just the most popular ones, L has been shipped with everyone in the show, and some characters who aren't.
Mello has been paired with quite a lot of people, from his heterosexual life partner (Matt), to his rival (Near), to a woman in whose shower he hid (Lidner), to the girl he kidnapped (Sayu), to people he's never actually met (Light), to just the downright creepy (Light's dad).
Near too. Both Matt and Mello, occasionally Linda, the SPK (Gevanni most of the time), Light, Mikami and L.
In Death Note-verse, Genius IQ = Launcher of a Thousand Ships. The three exceptions are A and BB, who despite being geniuses have been paired with three or four people at most, and Matsuda, who isn't a genius but has been paired up with everyone of importance in Death Note. Including old men and shinigami.
The cast must be having one big, hot-blooded orgy behind the scenes...even amongst the heterosexual options, for example, a notable solution to the Kamina/Yoko vs. Kittan/Yoko dilemma is to just let her have both of them. As far as solutions to Shipping Wars go, this fandom seems to take the route laden with the most kink and the least nitpicking.
The fandom somehow already figured that everybody is hot for Oz and/or Gilbert. Pairings like Oz/Gil, Alice/Oz, Shalon/Oz, Echo/Oz, Gil/Alice don't surprise anybody anymore, so the fandom went a step further. And so pairings like Oz/Oscar, Gil/Break, Gil/Vincent (it's actually almost canon - at least on Vincent's side; also, trope inverted, as Vincent is the one to obsess over his older brother Gil), Oz/Elliot, Gil/Jack, Gil/Oscar and Oz/Gil/Alice came to life.
At this point it's safe to say that everybody can be paired with anybody with how connected the plot is becoming. Luckily, the abyss can send people through time to quell most fandom issues of age difference if they want to be moral about everything.
In general, the author seems to try for this with the series overall. If two characters have spoken, there is almost always shipping material for them. (Some may need thicker Shipping Goggles than others, but you can really find a fic for anything anymore.) If there hasn't been enough, it's coming. Couldn't see where Gil/Break was coming from? Cue Gil leaving his beloved master to be Break's left eye. They don't even have to have had real interaction. Wondering where the fair-sized fandom for Break/Elliot could come from? The character guide makes specific note about the latter deeply admiring the prior.
Moreover, Break is easily at least as much a Launcher as Oz or Gil anymore, if not moreso. It isn't too uncommon to see Break paired with Sharon, Reim, Gil, Oz, Alice, Alyss, Cheshire, Vincent, Lotti, Elliot, Rufus, Shelly, Cheryl, or even Emily; and a few others are rare but also present.
The Darker than Black fandom has seen to shipping Hei with anything that has a pulse, up to and including the cat. Studio BONES did practically bait them into it — you know you've got a ship-magnet on your hands when he's written into slash fics in canon.
Despite it being the poster child for harem comedies, I'd have to say that Tenchi in Tenchi Muyo! ends up getting this treatment from fans. This is in a genre that named the Tenchi Solution and he STILL ends up with pairings outside it?
Of all the girls in Azumanga Daioh, it seems like Sakaki ends up taking this role.
Shinji Ikari has been shipped with every available member of the cast in an attempt by fans to make him happy by putting him in a stable, loving relationship, including Asuka, Rei, Misato, Kaworu, Hikari, Kaji, and Pen-Pen.
Most major character's can be shipped with almost any other character in the right situations. Rei can be shipped with Kaworu/Asuka/Gendo Ikari/Ritsuko although the last two are sorta...uhh...disturbing. Asuka can be shipped with Kaworu/Touji/Kensuke/Hikari/Misato/Kaji though the last two of those are also kinda awkward.
Keiichi has been paired with all of the girls: Rena, Mion, Satoko, Rika, Shion (despite the latter being in love with Satoshi), and occasionally even Hanyuu. He's also been paired with Satoshi and Irie. In one episode of Kira he ends up getting paired with Shion and Mion's Grandmother!
Izaya Orihara. It helps that he professes to be in a Mind Game Ship situation with the entire human race. Izaya/Shizuo seems to be the most popular, though Izaya/Mikado, Izaya/Masaomi, and Izaya/Namie also seem to have a degree of popularity, while Izaya/Anri, Izaya/Shinra, Izaya/Dotachin, Izaya/Rio, Izaya/Either or both of his sisters, Izaya/fem!Izaya, Izaya/Celty and Izaya/Celty's Severed Head (yes, those are two completely separate pairings) are not unknown.
There's also a small minority of people who ship Izaya/Prussia.
To be quite honest Erika and Walker are responsible for perpetuating a lot of the ships with their own otaku fangirl/boy squealings, particularly Shizaya.
Shizuo: I'm off to Shinjuku to kill Izaya...I'll be back
Erika: Oh man this is so cool! Shizu~chan totally has the hots for Izaya! Boy on boy love is so much fun, I love it!
Rest of Cast: NOT HAPPENING!
Mikado has his fair share of ships too. Despite his timid demeanour, he is paired with every cast member too. His canonical love interest, Anri, and his best friend ,Kida, are the two most prominent, but there is also a strong following for Mikado/Celty and Mikado/Izaya. The latter is especially focused on by readers of the light novels which also leads to a strong following of Mikado/Aoba and Mikado/Orihara Twins.
Puella Magi Madoka Magica: Madoka Kaname she's been shipped with Mami, Homura, Sayaka, her long-haired self from the opening animation... Just to put this into perspective: this is before even half the show's (12-episode) run has passed. Shippers work quickly nowadays. It's not particularly surprising or weird, given that Madoka/Sayaka's joked about in-show, Homura is very strongly implied to be in love with Madoka, and the ending is screwy enough for Madoka/Madoka to be Foe Yay.
Ciel himself has been shipped with many characters (the fact that he generates a lot of Hoyay with people like Sebastian, the Undertaker, Lau, Sohma, Snake, Arthur, Alois, Claude, McMillan, and Edgar most definitely helped with this). He's not a slacker in the heterosexual department either (usually Elizabeth, Ran-Mao, or Doll).
Tiger & Bunny fans are fond of these ships; Antonio/Ivan, Keith/Ivan, Pao-Lin/Ivan, Karina/Ivan, Barnaby/Ivan, Yuri/Ivan, Scarf-tan/Ivan...Guess which one of them is the launcher here?
Though Kotetsu is overwhelmingly paired with Barnaby, it isn't unusual to see people cop to shipping "Everyone->Kotetsu."
Blue Exorcist's Rin Okumura is frequently shipped with his twin Yukio, Bon, Shima and his brothers, Mephisto and Amaimon by Pixiv users. Incidentally he is voiced by Nobuhiko Okamoto who also voiced the aforementioned Ivan...