I have my OTPs, which are ones that I either think are sort of canon or could have been canon, and my challenge pairings, which is everything else.
For my OTPs I like having them in character, or as much in character as they could be, considering them being together would drastically change the canon in any way ( ie. John/Sherlock being together at the beginning of Series three). I also like vamplock/winglock/[insert spn thing here!lock] AUs, but only if both characters are of the type, or at the very least they are on equal footing.
The 'challenge' pairings are everything else, ones that will definitely never be canon but are fun to write. A great example would be Tim McGee/Tony DiNozzo, or Kirk/Spock/McCoy. I like these to be as in character as possible.
I don't really like Omegaverse type fics, but that is mostly because of the fact that it is often used as an excuse to write dub-con (which personally isn't my cup of tea for various reasons), but I've found one or two good ones.
I also have a massive soft spot for soulmate AUs...

Across multiple fictions, you will find many characters with too similar goals, or characteristics, or even desires. Having them interact, whether competing, cooperating, or even discussing it excites me. Them ending up in bed is just one way to climax the fanfic, so to speak.

I'm not really into romantic stuff, but some of the books I've read have had romantic subplots that fit in very nicely--not wedged in sideways, but growing naturally from who the characters are, and how they interact with each other. In these stories, romance is one of many things going on, and it serves a purpose in the plot, to show the reader more about who they are. Sometimes it creates complications when someone can't keep their wits about them because they're too in love; sometimes it means that the romantic relationship gets tangled up in the career or the quest and makes things more difficult. Sometimes it makes things easier because people who are in love can support each other and back each other up, just like close friends or siblings would.

I don't like stories that focus entirely on the relationship, as though that's the only thing going on in the characters' lives, like everything but the couple is either an antagonist trying to break them up, or else no more than scenery. That's ridiculous and unrealistic and, even without any particularly bad writing, is very likely to cause me to find something else to read. Romance in a living, changing environment, where it's only one of many things going on, though--that can be pretty cool.

Deranged plots and ships that make no sense and are often impossible due to space and time getting in the way but you don't care because it's SO DAMN GOOD AND/OR HILARIOUS.

What can I say? Crack's an addictive substance, kids. Someone should make it illegal. =]

I like fics where the characters stay in-character.

Fangirling warning

Right now I am really liking Duke/Nathan shipfics in Haven. They have this weirdly intense relationship in canon. They spend most of their time saying they hate each other and intentionally saying things to jab at the other's insecurities. But then, when one of them is hurt or in trouble, they drop everything and run to help.

In one early episode, Duke was dying of rapid aging, and Nathan (who is definitely the angrier party) had this entirely different tone of voice (gentle) and all these familiar little mannerisms and touches to comfort and help Duke through the day. (The touching is significant, because Nathan literally cannot feel touch, temperature, or pain, so touching is a very conscious thing for him. He doesn't touch people or allow them to touch him very often). And then there are these looks--especially from Duke--when he thinks no one is going to see them that are just pining. The UST between them is intense.

They have yet to reveal more than a few extremely sketchy details of their life together before the show started, but one thing they did say is they've known each other since they were 5, so they have 30 years of history to explore (and I am standing firm in my headcanon that they have at least slept together, and probably dated at some point), and I've about fallen in love with fics that cover that history. Even though I know whatever their history together, and at some point it had to have been good to give them that level of intensity of emotion, you just don't get that intense with people you don't care about, it obviously ended very badly, given their starting point in the show.

End of fangirling

Well, basically I love any fic in any genre and any fandom I read that covers backstory, it's just that those two obviously have so much backstory--both in time and significance.

I also like a good AU that shows people in a different life, but still being themselves. There is one about Clint Barton and Phil Coulson where Clint never left the circus, Phil never joined SHIELD and they still met and ended up together. Or one in that alpha/beta/omega dynamic--which I don't normally like, but this fic actually called it on all the horror that it would cause when applied to real people. Anyway, it was a Supernatural Cas/Dean fic with that alpha/beta/omega dynamic as the horror in the world instead of the demons and monsters that are the horror in the show. All the characters were absolutely spot-on in-character. All of the messed up family dynamics and history was there. It was not a nice fic, but it was very true to the show.

I like slice of life fics, fics where one party is comforting the other or they are both comforting each other, fics where they have to deal with unusual or unexpected circumstances, and fics about characters after they are done saving the world or whatever it is they do.

Okay. Now that I think about it, I read a lot of melancholy, depressing, violent, and sad stories. Stories that explore the limits of human endurance and the extremes of emotions and psychological resilience. I do like stories to end with hope though. I'm not sure what all that says about me. :P

Or pseudo-canon pairings, such as Sam/Jack from Stargate SG-1 (I have a particular weakness for this one). I say pseudo because it was obvious they liked each other, though they never got together.

I greatly enjoy stories that show on any level just how much a couple really love each other, canon or not. This often involves conflict to some degree, but I've seen a few good fics where they're just hanging out in the break room or nearest environmental equivalent.

And you can almost always get me with a story where one party gets in deep trouble and the other has to go and save them, although this particular story occurs as often in a friendship situation as a romantic one. Either's cool with me, as long as there's drama and adventure.

As for the fan goo, I love a story with a well-done Faramir/Éowyn element so much I could just cry. Except I don't, but I could. It doesn't even have to be the main point of the story, as long as it's there.

Fluff is always good for me.
Although, stories where there is a conflict (even minor ones) are also good, since it makes the couples more realistic. Not to mention the endings can make me turn into fan goo.

... and characters who make it work despite being kind of weird, which is probably why I stray into Holmes/Watson, Spirk, and Aziraphale/Crowley territory.

(Actually, I just love Good Omens, and I freaking love Crowley, probably because he just all-around sucks at being evil, not that Aziraphale is much more effective, given that it took both of them over a decade to figure out that they'd had their eye on the wrong kid...)

The best ship fics, in my opinion, go on to explore exactly what the consequences of said ship are, especially when those consequences are negative. I remember one fic that shipped Rarity and Pinkie Pie, and then went on to consider what this meant in regard to Rarity's career, as well as her realistically showing what her emotions about the whole relationship would be in the first place.

Fluff. I'm overly fond of fluffy oneshots (for established canon pairings, mostly, because I don't spend the fic going 'wait, what?'). The best ones obviously stay in character and aren't overly sappy. Actually, in one of my most favorite shipfics, the pair never interacted at all.

It was absolutely hilarious: It started off with Tonks filling out papers in the auror's office and a fellow female started asking her if she had any plans for the evening. Tonks had guard duty for the Order that night, but she obviously couldn't say that, so she said she had a date. Of course, her colleague immediately wanted to know more, and Tonks inadvertently began describing Remus when her colleague started asking for details. When the colleague wanders off to finish her own work, Tonks gets back to her papers and signs it- and then she realizes she signed it Nymphadora Lupin.

Fluff for couples who have a tragic ending make me ridiculously happy before I remember what happens to them and I promptly burst into tears. If a shipfic can make me do that, it's good to me. :3

I'm the biggest derp when you show me fics where Sherlock and John or Kirk and Spock are soulmates bound by red strings of fate or matching words on their fingers or TiMERs that hit zero together. It's the cutest thing. My interpretation of soulmate is just someone who, after they enter your life, change it for the better in some way that you can't imagine life without them, and that can be seen through a platonic or romantic light.

Oh, and Pacific Rim AUs where the pairing pilots a jaeger together. Because once again, mental soulmates.

While I am adverse to most fan pairings, the ones I do like are ones where the characters' tastes and personalities harmonize well.

Let's take the popular Rarity/Fancy Pants pairing for example. Both love high-class gatherings and fashions. Fancy Pants likes Rarity a lot, and his opinion her doesn't change after he finds out that she is from the small town of Ponyville. Rarity enjoys his company not only because he likes her, but also because while Jet Set and Upper Crust may sneer at the other Mane Five, Fancy Pants is a gentlecolt and treats her friends with respect. He doesn't car what class another pony is, and that makes Rarity like him all the more.
They are also foils for each other. She is very dramatic and excitable while Fancy Pants is cool, calm and collected. For me that is an ideal pairing.