What's non-beta version of Xcode works with iOS 13.6.1?

iOS 13.6.1 is just a dot release to the public, and should work with Xcode version 11.6 - run Xcode, connect that device, authorize it for dev work and then wait for Xcode to collect symbols. Then you should be able to select and build to it.

As for what with what, best to start with the release notes - example: https://developer.apple.com/documentation/xcode-release-notes/xcode-11_6-release-notes

I am having this same issue, was on Xcode 11.5, my device that updated to iOS 13.6.1 was flagged as unsupported, so I've updated to Xcode 11.7. I've tried cleaning the build folder, deleting intermediates folder, restarting Xcode and device, still stuck, can't debug on the phone. I did download newer simulators and Xcode is fine with 13.6 (and 13.7) on simulator. Any solution to get un-stuck on this error?

Exactly the same problem here. Was on XCode 11.5. Stopped working with iOS 11.6.1. Upgraded XCode to 11.7. Still doesn't work after trying a clean built and un-pairing/re-pairing. Haven't tried yet (installing XCode takes *forever*) but I guess using XCode 12 Beta may be the only solution(?).

What's weird though is that the "What's New" description of Xcode 11.7 states that it supports iOS 13.7. So you would expect that should include iOS 13.6.1???

Update: I never got it to work with iOS 13.6.1 but I noticed a new iOS version (13.7) has become available. I installed it on my iPhone 11 Pro and am now able to install my App using Xcode 11.7 (which, as I can tell, is not in Beta anymore).