I'd recommend you always use JPEGXR, it works great on all platforms, it's the best size/loading/artefact compromise.
But they are really big compared to PNGs, which is going to be a problem if you're targeting Android (APK has to be below 100 MB).
I experimented a bit with PVRTC/ETC/DXT. PVRTC is unusable. The quality is just a mess, don't even try it - a waste of time. Other two are not that bad, but you'll still see a quality drop. Lossless JPEGXR are a lot bigger than PNGs (with my project I get a 5.1 MB PNG and 5.8 JPEGXR - after compressing both with zip). With many atlases for different resolutions losing ~20% per image is a lot. Also JPEGXRs are loaded as ATFs so (I think) they won't use premultiplied alpha, which is not handled correctly in Starling (some things might not render properly with different blend modes).
WebP would be the best choice in my opinion, but they are not currenlty supported in AIR (you can use an ANE to load them, but this will be synchronous, so slower than PNGs). I created a feature request for adding support for WebPs on Adobe's bugtrack you can vote for: https://bugbase.adobe.com/index.cfm?event=bug&id=4130714 As soon as they add this it's going to be a no-brainer what image format to use.
TL;DR: Don't go with ATFs unless you really have to.