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

The answer, imho, is stable as it receives minor bug fixes and security patches without changing most of the code significantly, usually resulting in greater compatibility as you upgrade through each patch.

The downside is that moving a minor version increase to, say 3.3.0 (in the future) might involve more significant changes you will have to deal with in “one go” instead of as a drip feed. This can especially impact customisations, e.g. bespoke plugins, that were built with the prior minor version in mind, but not tested on the new 3.3.x series of beta releases.

On balance I’d suggest going stable if you don’t mind missing out on the very latest features.