Personally I tend to go AUR first, then Flatpak and then Appimage if there's no other choice. Snaps never lol

The reason being, I find that Flatpaks sometimes have issues with not being able to access certain things in the filesystem which can cause problems. That's presumably by design since they're sandboxed and you can fix it with Flatseal or whatever, but it's an extra level of fiddling that I can't always be bothered with. I do prefer Flatpaks for certain things that are messy with dependencies though (looking at you, Steam.) Appimages I don't really like because I hate having to go and check manually for updates for each one, it feels too much like Windows to me. But there are a couple of things that only have Appimage versions so I'll suck it up.

Snaps I just find to be a huge pain in the ass, and I've never found an app I need that doesn't already have a version on the AUR or as Flatpak or an Appimage, so I really have no need for them.