What’s the best/safest branch not break production site?

Word. And this is based on several years of experience in running a large-ish forum as self host. While beta or test-passed receives fixes and new features faster, the stable branch is predictable.

You know when major changes happen (version upgrades) and that in between versions the changes are minor. This way you can plan your own resources, prepare for some necessary testing effort, and potential breakage of your customizations (custom themes etc.).