I almost always ship canon. When it comes to why, I've never had a clear cut reason. Typically I'm shipping the couples I'm really invested in before they've become canon, but there's no doubt that me having a good idea that they're going to be canon makes me more receptive to them. I don't think I just blindly ship anything canon though. There are canon ships I like, canon ships I don't care about either way, and canon ships I dislike. But it's extremely rare that I ship something that isn't canon. It's just never been something I do.
My first ship that I knew didn't have a good chance of being canon was Tokka from Avatar: The Last Airbender. Now don't get me wrong, I hoped it would be canon, and I actually did think that it would play a bigger role in the show than it did (Toph did have a crush after all). But I was never one hundred percent certain that that was how the show was ending like I was with, say, Kataang.
Other than that, I don't have many non-canon ships even today. There are some fanon ships that I like well enough, and I enjoy reading fanfiction for them occasionally, but I don't seek it out. The only other major one I ship is Leah/Embry from Twilight, but that feels different from Tokka just because neither one of them ends up with anyone in canon. In my head, it's incredibly easy to imagine that they got together, but it just happened after the books ended.
If anyone reads my fanfiction, you know that I wrote a Gale/Johanna fic, and I'm actually quite happy with that fic. I think I'd get a mixed reaction telling readers of that fic that I don't entirely ship Gale/Johanna. I like them together, but I kind of shipped Gale/Madge throughout the books. Mockingjay was the first book where I considered Gale/Johanna, and I still didn't ship them as much as Gale/Madge. (And I really didn't actively ship Gale/Madge.) The reason I wrote that story (other than the fact it was a canon continuation and therefore couldn't be Gale/Madge) was because Johanna was my favorite character. I primarily wanted to write about her, and I saw her with Gale more than anyone else. I also strongly disliked Gale, so I thought writing him more would be interesting. I actually do like him better than I did before writing it. All of that being said, I still wouldn't consider myself a Gale/Johanna shipper. I just put them together because it made sense to do so to me, but I'm not at all invested in it outside of that one story.
Oh, I could bring up Makorra, but I really don't count that since they did date in the show.
I really admire people who ship couples that aren't canon with so much passion. That is incredibly impressive to me. It's just never been one of the responses I've had to fandom.