I carry on about my love for my primary ship, Xena/Gabrielle, fairly often, at least for my level of dreamwidth talkiness. So I'll just say a few things that I love about this ship.
The first reason is that it's femslash, which I tend to go for more than het or guyslash (with the general acknowledgement that I'm not super shippy generally, and also the hopefully obvious caveat that I'm not criticising het or guyslash, just saying that it doesn't usually hit my buttons). Another reason: the characters have great chemistry together.
But the bigger reasons have to do with the kind of epic love story that Xena: Warrior Princess tells. Xena and Gabrielle's relationship is built on the kind of trust that comes from years of fighting alongside each other, seeing the best and worst in each other, and changing each other.
I also love that, while things like jealousy and insecurity exist in their relationship, it's not the primary source of drama in it (they're much too good at communicating for that, plus, like I said: years of trust). Their drama mainly comes from the ways that Xena's past comes back to haunt them. From their need to constantly negotiate the tensions between their love for each other, what kind of people they want to be, and what kind of people the world requires them to be.
I'd love to find an ace reading of Xena/Gabrielle, that presents their relationship as a powerful, emotionally strong and deeply committed asexual one. I think that would be cool. I think some of the Xena/Gabrielle "friendshippers" are probably homophobes and some of them are probably coming from a perspective like that one.
Other ships I like:
Root/Shaw from Person of Interest, my only antagonistic ship. Those two have amazing chemistry. (I stopped watching mid-S3, but goodbyebird's cap posts of Root/Shaw sometimes make me want to start again).
Buffy/Spike, but ONLY in season 7, and I like it as much as a non-sexual partnership as anything else. Their "I've seen the best and worst of you and fought at your side" thing is somewhat similar to Xena/Gabrielle in its appeal, though to a much lesser degree for me.
Mulan/Aurora in Once Upon a Time. I haven't watched since the mid-season hiatus last year, so my most recent thoughts on this one are that they disappointed me, but for a while there they were the butch/femme ship of my heart.
Willow/Tara, in seasons 4 & 5.
M'Lila/Lao Ma, in an AU where they both live. My only ship that isn't canon or super strong subtext, and entirely born out of vidding Become You and realising they'd be awesome together.
I feel like I should note that Tammi/Eric in Friday Night Lights would be my ship, except their ongoing conflict about how they prioritise Eric's career over Tammi's hits me way too close to home. It's well-handled by the show, but their relationship is often painful for me to watch.*
All in all, I'm a pretty boring shipper. I don't like relationship dynamics that seem unhealthy to me, I'm generally not into hateships, I am mostly delighted by loving, supportive, long-term (would not need to be monogamous, but usually are) partnerships. Especially if they are between women. I think this is related to why I have so few ships. ;)
*For the record, this is not an issue in my current relationship. I am one of the very few women scientists I know with a male partner who prioritises his career after mine.