Let's start by listing the characters and pairings I love and do not love, and then I can start pulling out commonalities. Some of these canons could have much longer "love" lists; in those cases, I've limited myself to four of each.
Love: Rydia, Yang. Rydia/Edge, Yang/Yang's wife.
Do not love: Kain. Palom. Tellah. Palom/Porom.
Do not love: no one else stuck with me enough to really love or not-love them.
Haven't finished it yet, and therefore I don't have well-enough formed opinions on the subject
FFVII (OGC/AC only)
Love (well, like; this is the FF canon I am least interested in): Tifa, Aeris. Tifa/Cloud.
Do not love (like): Cloud, Sephiroth. Yuffie/Vincent, Cloud/Sephiroth, Aeris/Sephiroth.
Love: Quistis, Zell, Laguna, Rinoa. Rinoa/Squall, Quistis/Squall.
Do not love: Squall, Seifer, Selphie. Seifer/Squall, Laguna/Squall
Love: Freya, Vivi, Dagger, Beatrix. Dagger/Zidane, Beatrix/Steiner.
Do not love: Eiko, Kuja
Love: ...all of them. But based on my "limited to four" rule, I'll say Auron, Lulu, Kimahri, Braska. Tidus/Yuna, Rikku/Wakka, Auron/Lulu, Tidus/Rikku.
Do not love: Seymour. Auron/Rikku, Auron/Tidus, Wakka/Lulu, Seymour/Yuna.
Love: Paine, Nooj, Baralai, Yuna. Paine/Nooj, Baralai/Yuna.
Do not love: Rikku (in her FFX-2 incarnation), Brother (ditto). Baralai/Paine.
Love: Balthier, Ashe, Penelo, Larsa. Balthier/Ashe, Al-Cid/Ashe, Penelo/Vaan, Larsa/Penelo.
Do not love: Dr. Cid, Vayne, Venat. Balthier/Fran, Ashe/Vaan, Basch/Balthier, Larsa/Vayne.
Love: Fang, Lightning, Sazh, Cid Raines. Fang/Vanille, Fang/Lightning, Raines/Rosch.
Do not love: Snow. Lightning/Hope, Snow/Hope.
Looking over this list, some themes in the non-loved 'ships emerge right away. Incest is a big one, as are pairings that feature a distinct age difference and/or power differential. If by this, you came to the conclusion that Larsa/Vayne is my squickiest pairing in the history of ever, you would be correct! This extends to relationships between characters who seem, to me, to have more of a familial dynamic (Lighting/Hope, Snow/Hope, Wakka/Lulu) or friends/partners (Baralai/Paine, Balthier/Fran) than a romantic connection. I'm also not usually a fan of rival ships, particularly between the protagonist and their enemy.
The ships I do like? The first thing I usually need is some text. Interaction in canon. (With one obvious glaring exception -- Raines/Rosch -- which breaks a lot of other patterns, too. I'm really not sure where that one came from) Subtext alone almost never does it for me, although sometimes just a smidgeon of text is enough for me to open up to seeing subtextual cues as well. Because of this, and also because many of my favorite characters are female, I tend to gravitate towards het rather than slash.
As for characters, it's pretty clear here that I gravitate toward heroic characters rather than villains, but at least in the context of the Final Fantasy games, rarely is the protagonist my favorite character. My very favorites are usually just a step below the main hero: the support staff. Another type of character I like is one with a mystery. A character who has something hidden, just below the surface, but close enough that I can see that something is there, something worth discovering. One-note villains and "good" characters who don't show enough emotion for me to connect with them are typically the ones I don't care for.
The one exception in the character realm is the least-favorite FFX-2 characters, and that's all about the character development. I really liked the way Yuna's character developed and grew, showing more independence and while staing the same caring person beneath the perkier exterior. It felt very natural to me, the specific ways in which she changed and did not change. On the other hand, I didn't like the development on Rikku and Brother at all; it didn't feel true, or natural, and especially in the case of Rikku it was too bad, because I rather liked her in FFX.
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