I ship him and Shyvana. I think Shyv's a great balance for him. Even though he has that noble air about him, Jarvan also has a dark, all-consuming anger that sometimes gets the better of him; Shyv, conversely, comes from the domineering, primal dragon background, and has necessarily learned to master her rage. She's a great fit for him that way. Also, politics-wise, I don't think potentially having dragons on your side is ever really a bad thing.
Leona couldn't look past that darkness within J4 I don't think. As far as alliances go, I don't think the Solari give much of a shit one way or another; they stand alone. I'd rule her out.
Quinn doesn't gain Demacia much; I also see Quinn as a sort of stargazing girl who's grown to accept that it can't be, but still kind of wishes it would happen anyway.
Others...Lux? Meh. I feel like she looks up to J4 as a (more recently distant but always separated by rank and nature) childhood friend, so they don't really mesh that way.
Sejuani? Nah, that lady's bitter for all of the help she's never been given, least of all from the prosperous Demacian nation. She knows right where Jarvan can stick Bristle's tusk.
I don't see any of the Piltover ladies falling for him. Vi and Cait are from a different culture and social status, they'd never get along with J4. Jinx is nuts, so no.
Karma? I don't think she'd let herself get weighed down like that. Also, Ionia is pretty independent on the whole. Irelia goes the same way.
Riven doesn't offer Demacia anything but her blade, even if she might be an okay match for the Jarvster.
So yeah, in my head canon, Shyv makes the most sense.

