PostgreSQL 13 is still the officially supported version, with version 13.15 being released last month and still supported.
We have a good number of sites running version 15, and that is a known working version that we plan on shipping an update for self-hosted users eventually.
Version 16 is not widely tested outside of developer machines, but if you are feeling adventurous and want to give it a try and see if something breaks, let us know how it goes!