A union mount presents a combined view of multiple directories (branches) in a single hierarchy. Ok, but how do I use one in practice?

Many union mount technologies are available on a modern Linux system: OverlayFS, aufs, UnionFS, various FUSE-based filesystems, …

Given a use case, how do I decide which one(s) are suitable? In particular:

• Which ones require root access on a typical system?
• Which ones can be mounted on / (with root access of course)?
• Which ones support configurable policies for writes (i.e. not necessarily writing to the top branch)?
• Which ones support writing to the underlying branches?
• Which ones support a priority order between branches that depends on the files?

Usage examples would be appreciated.

Please avoid answers that focus on a specific point. I'm looking for comprehensive answers that review the available software (something similar to What is a bind mount?). Good answers will be bountied.