My idea of 'shippy' tends to be pretty limited. Partly that's because, in general, I'm just not as interested in romantic relationships - in fiction or RL - as I am in other kinds of relationships. See for example the Spike-Dawn and Sam-Dean sibling relationships and the Buffy-Faith BFF relationship (which mostly exists only in my head).
There's other kinds of intimacy than sexual/romantic intimacy, is what I'm saying, and when you've already got serious chemistry of another kind, I don't necessarily see it as an improvement to add a sexual/romantic component. See Spike and Dawn, for example: Y'all know I adore them together, but even with Buffy out of the picture I can't ship them; I love the relationship they already have too much. When two characters (people) are already so close, sex/romance isn't necessarily more or better. Sometimes, it's just different.
Which isn't to say I don't ever ship anybody. I ship lots of people. Honest! Thinking about it, I end up grouping most of my hardcore ships into three categories: sweet/functional, twisted/bad/wrong and complex/interesting. (It is possible this covers the whole gamut; I dunno. Do you guys have other flavors of shippiness?)
* Spike/Buffy - comes in all three flavors! However, I much prefer complex/interesting (the Barbverse, for example) and sweet/functional (all my happy post-NFA Spuffy peeps!).
* Helo/Athena - I fell in love with them for the complexity. I love relationships that cross boundaries (race, species, chronological era) and struggle with lots of Big Weird Relationship Issues, which these two have in spades. (But they also both really care about each other and take big risks for their relationship, and I love that, too.
* Gaius/Six. For the twistedbadwrong first of all, because how incredibly screwed up these two are! But then for the hope that they could conceivably salvage something functional out of this mess. Those may be my favorite ships of all - the ones that should be unmitigated disasters, and aren't. (See also: Spike/Buffy.)
* Wes/Illyria - Again, Big Weird Issues. Does it get any weirder than these two?
* Spike/Dru - Because they do twistedbadwrong and sweet/functional at the same time.
* Faith/Giles - This is my oddball ship. Out of the whole list, this one has by far the least canonical support (and what little there is of it is of dubious canonicity anyway). This is the one ship where I consciously sat down and said, these two are more interesting romantically than otherwise. They're so different (stuffy old Englishman vs. streetwise smartmouth Boston girl) and yet so much alike (scarybad backgrounds, struggles for redemption, emotional vulnerabilities shuttered and barred, hidden darkness, wounds from years of fighting these supernatural battles, tattoos). She could shake him out of his funks; he could stabilize her frantic insecurities. They could be so good for each other. And also? Smokin' hot together.
* Zoe/Wash - Adorable and functional! Their love is open and direct, they're comfortable with each other, they're sexy together, they can laugh at each other, they're a bit quirky, they're committed, they're mature adults. Maybe most importantly, as important as they clearly are to each other, they don't define one another. Wash was a pilot before he met Zoe, and he still is. She was a soldier first, and she still is. They each continue to have important relationships outside the marriage.
* Kitty Charing / Freddy Standen (Cotillion, by Georgette Heyer) - Okay, yes, random book!ship, but I love these two so much. I don't think I have ever shipped any book pairing like I ship these two, because they are so perfect for each other. Freddy may be my favorite comedic hero ever - he's no intellectual and he cares ten times as much about clothes as I do, but he's also kind and gallant and thinks of others first, and on top of it all he is so wonderfully, unintentionally funny. And Kitty is warm-hearted and just as joyously frivolous as he is, and they end up liking each other so much despite themselves. I adore them.
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