Skye definetly takes the cake. There is Skye/Ward, Skye/Lincoln, Skye/Simmons, Skye/Fitz, Skye/May, Skye/Coulson, Skye/Tripp, Skye/Raina, Skye/Robbie, Skye/Mack, Skye/HIVE, Skye/pretty much every side-character you can think of, Skye/every Avenger no matter if old or new, Skye/Bucky, Skye/Rumlow...
Fan discussion around the show and shipping, especially post-Season 1, tends not so much to ask "Which couple do you ship?", rather "Who do you ship Simmons with?". In roughly descending order, the top five seems to be Fitz, Ward, Skye, May and Triplett — leaving only Coulson omitted out of the main cast. The former three also seem to be the most popular ships in the fandom overall.
Ward isn't far behind. In addition to actually hooking up with May and having feelings for Skye, he's shipped with Fitz (generally with a Seme and Uke dynamic) just as often as he is with Simmons.
The introduction of Mockingbird in Season 2 opened up the floodgates to Bobbi Morse being shipped with most of the other leads, as well as several other characters from elsewhere in the MCU. The favourites are usually Hunter (her canonical love interest), Hawkeye (her love interest from the comics), and Simmons (because the two enjoyed some serious Les Yay in their first couple of episodes). However, like Simmons, Bobbi will generally get shipped with anyone else on the team except Coulson, and there's a fair portion of the fandom shipping her with Skye, May, and Ward as well, despite the relative lack of interaction with them, just because she's so well-liked in general.
Fitz has been getting in on it too. Not only is he shipped with Simmons and Ward, but come Season 2, his budding friendship with Mack is causing many fans to hope for a little Bi the Way action.
Among the corner of the Battlestar Galactica fandom that has special love for Lt. Felix Gaeta, he has been happily paired with every character (male, female, human, Cylon) you can think of. Including quite a few he has no real interaction with on the show.
And there's no possible way of counting the Kara "Starbuck" Thrace pairings. Or threesomes. Or… that was bound to happen because in the show itself she Really Gets Around. Gaius Baltar and Ellen Tigh also have a number of pairings. Even Lee has a collection.
Faith Lehane from Buffy the Vampire Slayer is a Launcher of a Thousand Ships not just for her own fandom, but for many, many others. Just in fandom there's been Faith/Cordelia, Faith/Dawn, Faith/Buffy, Faith/Willow, Faith/Dawn/Buffy/Willow, Faith/Spike, Faith/Angel, Faith/Wesley, Faith/Xander, Faith/Tara (who she met once, in Buffy's body), Faith/Tara/Willow, Faith/Robin Wood (canon, but they broke up between Seasons 7 and 8), Faith/Giles, Faith/Joyce, Faith/Anya, Faith/Harmony, Faith/Lilah Morgan (admittedly, Faith did hit on her the one time they met) and a few Faith/Everyone fics. And that's just in fandom.
Spike is another lucky one that has been paired endlessly with every imaginable character, including ones he never met. Like Ampata, the Inca mummy girl, who was only in one episode. Spike wasn't in it.
Angel is another, though it's more understandable, given there are the ships from Buffy and then the ships from Angel. There's canon Angel/Darla, Angel/Dru and Angel/Buffy, and then there are Angel/Spike, Angel/Wes, Angel/Lilah, Angel/Fred, Angel/Lindsey, Angel/Cordelia. Gunn is possibly the odd one out.
Buffy even gets her share. Outside of her canon love interests Angel, Riley and Spike, there's Buffy/Willow, Buffy/Xander, Buffy/Dawn, Buffy/Anya, Giles/Buffy, and the ever so popular Buffy/Faith. To name the characters she has shipped with both in and outside her own series would take up the whole page.
Criminal Minds' Reid. Fandom ships him with everyone. Even half the unsubs. And there are also a lot of crossover ships.
Annie Edison is this in Community, evidently. About half of the speculations have been hinted at in canon.
More than half really. She's had (some admittedly small amounts, but still) Ship Tease with Jeff, Britta, Abed, Troy and a canon relationship with Vaughn (along with multiple minor characters, such as Starburns and Professor British Guy pointing out how smoking hot she is). That really only leaves Shirley and Pierce.
There's a further irony regarding Annie's status as the show's resident Launcher of a Thousand Ships in that she's also clearly established as being the most prudish, repressed and naive of the characters concerning matters of sexuality, and tends to stumble into such matters unwittingly.
The episode featuring the Crack Pairing description features pairing of most possible characters. Abed is very into every possible pairing.
Pretty much every main credits character in the CSI franchise series is this by now. It's easier to name the ones who *haven't* been paired with everyone else in their own show and several in the other shows.
Degrassi has Sean. It doesn't help that every girl he's shared a scene with (and a few he hasn't) have said how hot he is. For reasoning, Emma, Ellie, and Ashley are canon, Manny talks about how hot he is, Paige and Hazel are popular Crack Pairings, and that really leaves Liberty, which goes to show accidents will happen. For Ho Yay, let's see… He's the only person who truly understands Rick, he and Ellie defend Rick before anyone else does, and then he kills Rick; on TV.com, this combines with speculation about Toby's taste in anime to turn the yaoi up some. Then there's Craig; indeed, Sean/Craig is the official couple over at TV.com's Degrassi forum, complete with a user named Seaig. More relevant, though, is that Craig wanted Sean to run away with him, and attempted suicide when Sean didn't. For Jimmy, there's this Foe Yay between them at first that is Serious Business, and then Jimmy's girlfriend cheats with Sean and goes on and on about how Sean's the better man, sprinkling in double entendres throughout. JT and Toby think Sean scores a lot, and JT tries to get him to use X. Marco's first crush is actually Sean, complete with Male Gaze shot of Sean's naked torso. Jay is a popular choice because of how mopey Jay gets when Sean isn't around. Peter has a serious obsession with Sean during their Foe Yay episodes. As for Spinner? There's actually a promo pic of Daniel Clark embracing Shane Kippel.
Not even Liberty was immune to it. In Season 3, she had a very blatant — if only for one episode — crush on Sean. On top of that, there's Ashley, too, which is sort of Canon.
For slash-alone, Marco. Some fans just aren't that original.
Eli seems to have inherited Sean's throne. Canonically, he's been with just Clare and Julia, with hints of him and Imogen becoming an item currently there. However, within the fandom, he's shipped with everyone. Special mention goes to Fitz, Adam, and Fiona, however. The former of whom is a (As far as we've seen) straight male, as well as his rival during Season 10. And with the latter two Adam is an FTM Straight Transgender, and Fiona a lesbian. And then there's Eli/Morty the Hearse…
Jack Harkness of Doctor Who and Torchwood. Of course, he'd almost certainly flirt with everyone listed on this page. There is a Image Macro series out there pairing him with (among others) CJ Cregg, of all people.
This one has both been artificially inflated and taken to the next level by a fic challenge on a very popular Doctor Who community, called "Jack Harkness Sexes Everyone in Every Fandom Ever", which has at its core the absolutely impossible goal of writing enough fics to achieve just that.
The Doctor himself. Especially since the newer Doctors — Eight, Nine, Ten and especially Eleven — are a bit on the slutty side, and the Doctor has been written as bisexual ever since the Eighth Doctor Adventures. While the Doctor/Rose, Doctor/River, Doctor/Clara, Doctor/Fitz and Doctor/Charley shippers are the most vocal (and have the most canonical ships), there's strong support for the Doctor/Martha (who kiss), the Doctor/Jack (who kiss), the Doctor/the Master (who kiss), the Doctor/the TARDIS (who kiss), and this is all just from the new series. Even from the old series, there's the Doctor/Romana (he kissed the first in a non-canon special, and the second in the books, and the actors were married in real life), the Doctor/Sarah Jane Smith, and many, many more. Really, you can just say Doctor/X Companion and there's probably someone shipping it.
The Brigadier, because of his insanely long tenure, can logically be paired with each of the first seven Doctors (Eight, if you count Big Finish) in turn without any stretch in continuity. Season 7 of the new series actually made a deliberate Ship Sinking effort for the pairing, but as we know, the Doctor lies.
If you give the First Doctor Era fans more than a passing glance, you'll find that Steven gets shipped with just about every character he comes into contact with including Dodo, Vicki, Katarina, Oliver, Sara Kingdom and the Doctor himself. it certainly helps that the Doctor Who Expanded Universe makes it clear that Steven's from a Free-Love Future.
Clara Oswin Oswald can, canonically as of "The Name Of The Doctor", be shipped with every single Doctor Who character ever. If her personality doesn't match the ship — no problem, this Clara can just be a bit different. If she's shipped with the Doctor but he's not supposed to remember her face before "The Snowmen" — no problem, he gets Trauma-Induced Amnesia every few seasons anyway. The Clara/Ace ship on Ice World, in particular, has lots of leather jackets and fabulous '80s Hair.
Jo Grant, from the Third Doctor era, lends herself extremely well to shipping. She went on dates with Mike Yates in the Expanded Universe, married the leader of a hippie commune in the TV series, and has met up with the Eighth Doctor (in the novels) and the Eleventh (in The Sarah Jane Adventures). Add to that the picture of actress Katy Manning posing naked with a Dalek, which was canonised in the IDW comic "Prisoners Of Time", as well as pictures like these with Twelfth Doctor Peter Capaldi…
And then there's Iris Wildthyme, a big old slutty drunken ham of an alleged Time Lady who claims to have "had" the Doctor and who fancies all his incarnations. In addition to the slew of lovers (men and women alike) and husbands who show up in her own series.
John Gage on Emergency!. The most common are Johnny/original female and Johnny/Roy, but he's also been paired with nearly every other major character and one or two recurring characters on the show by someone at some point. The only exceptions seem to be Dr.Early, Dr.Morton and Captain Stanley.
Firefly: Just every damn character. Pick any combination of the show's eleven regulars, and it won't be too hard to find a fanfic pairing them up.
It gets Jayne Cobb shipped with nearly everyone. Most common shippings include River or Zoe, but there's also plenty of fic involving Kaylee or Inara, as well as ones involving Simon or Mal. Jayne is the token barbarian of the series — he likes guns, booze and women, and he'll take any that seem available.
River gets paired up with a lot of people. The most common ones are Simon (which is nicknamed Crazy Space Incest), the aforementioned Jayne and Mal, but also with Kaylee, Inara, Wash, Book, Jubal Early, the Operative and even Badger. The fact that she fits this trope despite being seventeen on a ship whose median age seems to be about thirty is especially disturbing.
Rachel from Friends. Helped by the fact she was roomates with Monica (up until early season 6), Phoebe (season 6), Joey (seasons 6 to 10, intermittently) and Ross (briefly during 8 and 9). Several subtle (and not so subtle) ship teasing between her and the other girls (notably in some fights and "what if?" instances), some with Chandler (especially in early seasons) and she dated the remaining regulars. The character also, at some point, kissed every other person in the main cast, including the girls.
Brienne. She's been shipped primarily with Jaime and Tormund, but also with Podrick and Sandor. She was actually in love with Renly, is all but confirmed to be in love with Jaime and Tormund is completely besotted with her.
Kurt isn't quite as popular because he mostly doesn't get shipped with girls, but he has been shipped with EVERY male character to appear on the show, and as the series continues there's been more and more Kurt/Various Girls fic. Apparently fangirls like a challenge almost as much as they like canon gayness.
Blaine is also quickly becoming this — and unlike Kurt, he did have an episode where he briefly doubted his sexuality, which fans have taken as an A-OK to ship him with girls. He's still mostly been shipped with guys, though, but like Kurt, it's with just about every guy on the show.
Quinn is quickly becoming this, too. The fact that she's hopped from boyfriend to boyfriend in canon — and, as of Season 4, even had a Bi the Way moment with Santana — means there's more than enough establishment for it.
Sam also frequently gets this treatment, largely because of fans who keep holding out hope for Bi the Way with him, even though he's only dated girls on the show.
Former bully Dave has been shipped with Kurt, Sebastian, and Blaine, and a few of the straight boys. Most of this is because he hasn't appeared since the Bittersweet Ending to his gayngst storyline and many fans simply want him to be happy with someone.
In addition to Peter Petrelli, Sylar is also a Launcher Of A Thousand Ships in the Heroes fandom. Check out this LJ Heroes community: Rare Heroes. Look at the pairing tags list; Peter and Sylar have the most pairings. Peter actually comes in as a close second to Sylar, when you count them up.
Every Heroes fanfic is about Peter, Sylar, Claire, or Elle, usually a combination of two… or more.
This Livejournal, "The Heroes Slash Manual", applies this quality to Kaito Nakamura… after a fashion.
Duncan Macleod in Highlander. Aside from the canon Duncan/Tessa and Duncan/Amanda, there's a lot of Duncan/Methos, some Duncan/Joe and even some Duncan/Richie now and then. And Connor has likely been paired with him a time or two as well.
It seems agreed on by much of the House of Anubis fandom that Patricia makes a good pairing with virtually every other student in the house, regardless of gender. (And one teacher.) While they can be platonic most of them are viewed as romantic.
Barney in How I Met Your Mother. The most popular pairing is Barney/Robin, but it is by no means the only one.
Barney/Lily is HIMYM's biggest UC ship. He's also shipped with Marshall and Ted. Or Stella and Victoria. Plus the crossovers...
As it's his show, House obviously gets treated like the fandom bicycle by fic writers. He's been paired up with everyone, from Wilson, Cuddy, Chase, Foreman and Cameron to Vogler, Tritter and, um, a giant squid. Which clearly hasn't been on the show but still…
Chase too. I've seen Chase/House, Chase/Foreman, Chase/Cameron, Chase/Stacy, Chase/Wilson, Chase/Cuddy, Chase/Kutner and Chase/Lucas.
Sam in iCarly. She has fanbases (although small in some cases) for pairings with Freddie, Carly, Spencer, Gibby, Griffin, Missy, Principal Franklin, Wendy, Brad, Shelby, Jade from Victorious and Alex from Wizards of Waverly Place. The prevalent fandom interpretation of her is that she is bi.
The male version is Freddie. He has been paired with Carly, Sam, Spencer, Gibby, Wendy, Melanie, Shelby, along with several girls of the week, and two reasonably popular Crossover Ship pairings, one with Megan from Drake & Josh, and the more recent one with Cat from Victorious.
It's a Law & Order, so it isn't that obvious, but in the (admittedly small) L&O:SVU fandom, Olivia Benson is this. Probably because she's Mariska flippin' Hargitay.
On Lost Claire Littleton tends to get paired with anyone she has any sort of contact with. There’s Claire/Locke, Claire/Sawyer, Claire/Ethan, Claire/Boone, Claire/Hurley, Claire/Eko and Claire/Shannon from earlier Seasons and Claire/Miles, Claire/Richard, Claire/Jacob, Claire/Man in Black, Claire/Sayid, Claire/Kate, Claire/Juliet and yes Claire/Jack brought on by later ones.
Hawkeye Pierce from M*A*S*H has been paired with virtually every other character in the series, though the most popular pairings are BJ/Hawkeye, Margaret/Hawkeye (canon for an episode or so, actually), Trapper/Hawkeye, and Radar/Hawkeye. Justified in that Hawkeye tends to flirt with Anything That Moves, though he (apparently) only flirts with men to make them uncomfortable.
Morgana from Merlin (2008). The first Season inspired Morgana/Gwen, Morgana/Uther, Morgana/Arthur as well as some Morgana/Merlin, Morgana/(older)Mordred and Morgana/Nimueh, even though those two never even met! Season Two cut it down a little on the Morgana/Uther and Morgana/Arthur front but also inspired Morgana/Morgause (even though they're half-sisters) and (thanks to some fanmade vids for them) also Morgana/Sir Leon.
Ironic considering Morgana is part of no Official Couple on the show. Seriously though, in this fandom, everyone gets shipped with everyone. In a cast this pretty, they're all Little Black Dresses.
Morgana/Mordred!? But—but—but… in original canon he's…
Merlin himself is the fandom bicycle. He is basically shipped with every single character on the show. Canonically, there's Merlin/Freya (and, extremely briefly at the very start of the series, Merlin/Gwen). But add in the vast majority who ship Merlin/Arthur, another segment who ship Merlin/Morgana, some who ship Merlin/Sefa, Merlin/Nimueh, Merlin/Mithian (due to Ship Tease photos), a few who even ship Merlin/Uther (yes, it exists!), the growing popularity of the Merlin/adult!Mordred ship, and finish it off with the Ho Yay-tastic possibilities of Merlin/Lancelot, Merlin/Percival (especially after Percival carried Merlin in 4x02), Merlin/Will, Merlin/Daegal, Merlin/Gilli, Merlin/Gwaine and Merlin/Knights, and you've definitely got this trope. But, as mentioned above, this show itself is this trope embodied. Merlin is also popular in crossover shipping.
Gwaine is shipped with anyone and anything, as illustratedhere. In addition to pairing him with various main characters, Gwaine/apples is also popular.
Ruby/Red is also shipped with quite a few characters—Emma (as mentioned above), Archie (due to both characters' popularity with fans and lack of canon love interests), Gold (especially before the introduction of his canon love interest Belle), Snow White, and Graham by fans who like Graham's character too much to take the convenient Killed Off for Real exit to further their preferred Emma ship. Jefferson and Regina, too, even though they (and Gold and Archie, come to think of it) have never actually interacted with Ruby. August and Ruby had one conversation of Ship Tease and gained a small-but-loyal fanbase. And then there are those who liked Red's original love interest Peter from her tragic backstory… Red is also commonly paired with Belle and Frankenstein, though both pairings have certainly had the Ship Tease to warrant such followings.
Captain Hook. He was being shipped with Emma, Aurora, Ruby, and basically half the cast before his debut episode had even aired. Hook is even shipped with characters before they were added to the show (Ariel and Wendy, for example).
Many characters from One Tree Hill, but especially Brooke. Other than her canon pairings, including the one night stands (Lucas, Nathan, Chase, Owen, Julian, Chris Keller), she has been paired with Mouth, Jake, Clay, Peyton, Rachel and many others. She's also very popular with crossover shipping.
In Power Rangers, Tommy got paired with an inordinate amount of Mary Sues over the years, but Sky from S.P.D. really takes the cake here, ending up with every single one of his teammates in fic (all at the same time on occasion), and some of his enemies.
Speaking of Tommy, it's easy to see Tommy with Jason (classic), Goldar (obsessed with Tommy), Zedd (even more obsessed than Goldar), Rita (come on, if you were her, you would), Kim (canon, and it's been said that 45% of Power Rangers fics are slash and 54% are Tommy/Kim), his clone, Kat (canon), and Adam (the two most popular rangers ever). There are also more bizarre fics pairing him with Zack, Trini, and Rocky, to say nothing of Observe The Viewing Globe. Fast forward to Dino Thunder, and he's paired with every other cast member: Hayley (despite the inherent Scrappiness of dating Tommy in canon and not being a Pink Ranger), Anton (they sound like lovers), Mesogog (personal space issues), Elsa (fakes being in love with him), Conner (because heavy is the head that wears the crown), Ethan (apparently on the same grounds as Trini/Billy, that one can understand the other's technobabble), Kira (far too much), and Trent (because they were both evil). There's also a Memetic Mutation that sleeping with Tommy is one of the fringe benefits of being a Pink Ranger. No doubt someone's paired him with Andros because they look so much alike. We're just lucky David never became a ranger, or we wouldn't have had to wait until Ninja Storm for incest slash to come out.
Other male Mighty Morphin' Rangers have their fair share of shipping fanfics, notably Billy since he is considered the Lady's Man of the team albeit being a nerd. Alongside pair-ups of him and his crushes in the show, some of the favorite fan ships of him include: Billy/Kimberly (affectionately labeled Kimbilly or Billery by fans), Billy/Trini, Billy/Aisha, and Billy/Kat.
A sizable portion of the fandom has most emphatically thrown Ben Wyatt into this role. While he and Leslie are the show's Official Couple (and a highly beloved one to boot), there's also fic for Ben/April (and in some cases, combining them with April's canon husband Andy for a One True Threesome), Ben/Chris, Ben/Ann, Ben/Ron... someone out there probably ships him with Jerry (even if only for the pun).
Leslie herself has been paired off with Ann (due to the huge heaps of Les Yay between them), Andy, Ron, April, and in the case of Crossover Ships, Trevor from The Good Place (due to him sharing the same actor as her canon husband Ben), the Tenth Doctor, and Liz Lemon (due to the close real life friendship between their actresses).
Emily from Pretty Little Liars seems to get paired up with a lot of people: Ali, Spencer, Aria, Hanna, Maya, Paige, Samara, even Toby (despite being lesbian.)
Like Firefly this is probably due in part to Cameron being played by Summer Glau.
Sherlock Holmes from, well, Sherlock. The Fan-Preferred Couple appears to be him with John, but he's also paired with Molly (who canonically has a one-sided crush on Sherlock), Irene Adler (a lesbian until Moffat decided to contradict his earlier 'not wired that way' anti-John/Sherlock statement), Anderson and Donovan (both of whom hate Sherlock), Lestrade, his own brother Mycroft (whom Sherlock hates), and nemesis Moriarty. Season three added Mary Morstan (John's wife, making for an OT3 situation) and Charles Augustus Milverton (the season's Big Bad).
Clark Kent has been paired with every character on the show. Clark and Lex being the most popular Slash Couple rivaled with Oliver/Clark, then you have your Clark/Lois Shippers, your Clark/Chloe Shippers and a smaller number of Clark/Lana shippers (whose numbers dwindled with each passing season as more fans grew irritated with Lana). They even go far enough to pair him withKara or Jor-El. The writers even would paint him as a Chick Magnet on the show, even though he hasn't caught onto it.
Tess Mercer, an original character who is partly based on Mercy Graves, Miss Tessmacher and the original version of Lena Luthor, the one who was Lex's sister in the Silver Age comics, has been shipped with just about everybody on the show. This is helped by the fact that the showrunners seem happy to intentionally fuel this with regards to Tess, by having her flirt with just about everybody in the main cast, plus several guest characters.
Chloe Sullivan also served as a Launcher Of A Thousand Ships. The fact that many of her fans seem to use her as their own personal Audience Surrogate means that Chloe hooking up with virtually any attractive character can be seen as wish-fulfillment. To that end, the fandom has shipped Chloe with Clark (Chlark; there are also pairings of her and Kal and Bizarro), Oliver (a ship so popular that the showrunners themselves became Chlollie shippers in the last couple seasons), Lex (Chlex enjoyed a brief but dramatic spike in fanfic popularity around Season 2 for some reason), LIONEL of all peoplenote They actually had a bit of Word of God-confirmed Foe Yay in season four, and out-of-universe characters from other shows, like Dean and/or Sam Winchester from Supernatural.
One of the more popular Chloe crossover ships over the years has been Chloe/Batman. There have been so many prominent fics that it can be quite reasonably stated there is an entire sub-genre of Smallville fanfic that involves Chloe moving to Gotham (often after the writer has Chloe realize that Clark/Oliver/Jimmy/Lana/Lionel "isn't good enough for her" or something) and then falling in love with Bruce Wayne, and ending up fighting crime alongside him on the Gotham City rooftops. In these fics, she often displays the ability to work the Batcomputer better than Bruce can, learns martial arts at a ridiculously quick pace, and often Alfred will advise Bruce to "open his eyes and see what's right in front of him." This pairing is apparently helped by the fact that Chloe more or less becomes an Oracle Expy in later seasons due to Flanderization of her hacking skills.
Oliver Queen, aka Green Arrow, is subject to this a lot. Helped by the fact that the show actually does have him embark on relationships with most of the main female characters during his years on the show.
Rodney McKay, of all people. People will read anything that features him as part of a 'ship. If you watch the episode "Tao of Rodney", this makes perfect sense.
Elizabeth Weir: Rodney, you're a good person. Know that we love you.
Rodney McKay: You love me? Really? All of you?
Evan Lorne also fulfils this trope.
So does John Sheppard. More John than even Rodney, since there seems to be the subfandom of Richard Woolsey/John Sheppard. Also, John has spent time with every Atlantis Civilian Head in a 'one on one' scene on a balcony somewhere, and has DARED to fall for an ascended lady (and a near-ascended one). Poor John's reputation is also cemented because Rodney just doesn't let it go. In extreme cases, John's character kind of goes through a transformation, and people forget that he's actually the military head of the Atlantis expedition. John/Elizabeth, John/Teyla, John/Ronon, John/Rodney, John/Richard, John/Random Evil Alien, John/Random Good Alien, John/his abusive dad, John/the entirety of a hockey team… well. It's been done.
Daniel Jackson & Cameron Mitchell from SG-1 have been paired up in fanfic not only with each other, but with every main character in the show's history. Sometimes crossing over with Stargate Atlantis and other shows. Can you blame the other main characters?
Her lord, Takeru, has been paired with all five of his vassals, his childhood friend, his rival, and the true lord, who is now his mother.
Dean Winchester from Supernatural. He's been paired with his little brother, his father, Bobby, Castiel, probably every single male and female character in the show, OMCs, OFCs and I'm not even touching the millions of crossovers that he's been in. Possibly justified because he's very pretty and very slutty but it's getting ridiculous.
Castiel may count too. It started harmlessly with just Dean, but then other angels (and demons and humans) appeared, and the rest is history.
Crowley. It helps that he flirts with most everyone he meets.
The Suite Life on Deck features Zack Martin, a serial dater who seems to be compatible which every girl on the show.
Malcolm Reed of Star Trek: Enterprise. There is a significant amount of fic for: Tucker/Reed, Archer/Reed, Reed/Sato, Reed/Mayweather, and Reed/Hayes. No other character comes close. The reason why can only be guessed at, but it probably has something to do with the whole "shy, reserved, and stoic" thing. And his fondness for making things go boom.
In Sanctuary, Helen Magnus is paired through fanfic with everyone, including non-canon relationships with Amelia Earhart and H.G. Wells, among many others. Many of the popular ships in fandom include Magnus/John Druitt, Magnus/Nikola Tesla, Magnus/James Watson, Magnus/Will Zimmerman, Magnus/Kate, Magnus/… you get the picture. In canon, Tesla has a long-lived and one-sided flirtation with Magnus, but has gone so far as to sneak into her room at night (unsuccessfully) and has made it clear he would enjoy a relationship with Magnus if given the opportunity. Given that the woman is over 157 years old, is sexuallyfluid, and would have chemistry with a flea, it's not hard to see how this trope applies.
Supergirl: Kara Danvers has been shipped with virtually everyone, regardless of age, gender or universe: her co-worker Jimmy, her another co-worker Winn, her boyfriend Mon-El, her boss Cat Grant, her friend Lena, her sister Alex, her friend from another universe Barry Allen...
Stiles has been paired with pretty much every single girl (and guy, mainly due to his Ambiguously Bi moments) on the show, whether he interacted with him/her or not. And let's not get started with the crossovers…
Interesting thing on The Tudors. Henry gets paired off the most times in the show (obviously) but as for pairings written or suggested… um, Katherine of Aragon. A popular theory is that people just want her to be happy, considering how things went for her on the show.
Sookie from True Blood. Not only has she dated or had sexual tension with four guys canonically, but fan fic writers don't let silly things like gender or Incompatible Orientation keep them from pairing her with everyone on the show.
Sam and Jason as well. Both are tied in having the most onscreen love interests on the show (five, not counting Jason's random floozies), as well as some Ho Yay with a few male characters, making them very popular with shippers.
The Vampire Diaries: All of the characters in the series fall into this, especially Caroline (who has been paired with everyone even in-universe), the Salvatore brothers, and obviously Elena.
Cat in Victorious, like iCarly's Sam, is considered bisexual by the majority of the fandom. She has popular pairings with Jade, Robbie and Tori, semi-popular pairings with Trina, Beck and Andre and a popular Crossover Ship with Freddie from iCarly.
Jade is the secondary launcher, being paired with Tori, Cat and Beck as her most popular pairings, with less popular pairings with Andre and Robbie and a Crossover Ship with Sam from iCarly. Again, like Cat, most people assume that she is bisexual and that Beck has come to terms with her sexual needs as part of their canon relationship.
Alex in Wizards of Waverly Place. Goes above and beyond the usual fandom bicycle because her most popular pairing is an incest pairing with brother Justin.
Both Mulder and Scully. There are still whole websites devoted to shipping either of them with half the show's characters.
Alex Krycek. Do a little fanfic digging and you'll find him paired with almost everyone from the show. Krycek/Mulder is most common, but his level of Foe Yay with practically everybody spun off all sorts of other pairings. It's probably impossible to find a fic that doesn't mention his Green Eyes at least once.
Oliver Queen is shipped just about everyone in canon, while in fanfic he's shipped with anyone the writers don't tease him with. As of the second season, he's slept with/dated Huntress, McKenna Hall, Shado, Isobel Rochev, and both Laurel and Sara Lance, and has been teased heavily with Felicity, and less heavily Diggle, Tommy, and Malcolm and Slade. In the fandom, he's shipped with all of the above, plus more!
Not to mention Felicity: While her most popular ship is undoubtedly Felicity/Oliver, there's also Felicity/Sara, Felicity/Diggle, Felicity/Walter, Felicity/Laurel... and her canon Ship Tease pairings Felicity/Barry and Felicity/Ray.