I love shipping actually! It's cute and fun. And also, I'm still internally 12. Sorry this is kinda long I just get REALLY excited over shipping.
I love Pharmercy to death. It's what got me into Overwatch, actually! And what stemmed my battlenet username. I have a HUGE emotional attachment to that pairing, whoops.
Zarmei is cute, too (Zarya x Mei), Anahardt (Ana x Reinhardt), Symbra (Sombra x Symmetra). McHanzo (Hanzo x McCree), and Lenily (Tracer x Emily, which is already canon! Yay!). I have a few others I like, too, but those are just ones I can name from the top of my head.
And ooh, Gency. I hate Gency with my LIFE and I could go on a really , REALLY REALLY long rant as to why but that's for another day...There's so many reasons why it's awful. Blegh.
I hope nothing ever becomes canon (it probably won't since Jeff himself said he'd leave that to the fans, thank goodness), but shipping makes up a big portion of the fanbase and has even stemmed a few voicelines or nods from devs, which is super cool.