Badge choices in England:
County
FAMOA
3 Lions
Individual league
For the FA Cup, you are required to wear your County badge, unless you have been awarded the 3 Lions. You get 3 Lions if you officiate (middle or line) an FA final (FA Cup, Trophy, Vase, Women's FA Cup, Youth FA Cup etc) or if you are awarded your FIFA badge.
For FA Cup, this is the order (and only acceptable choices) for badge wearing.
FIFA
3 Lions
County.
For Contrib, you are required to wear a shirt with a FAMOA badge on, or that of the individual league.
At Supply level, you are expected, but not required, to wear a a league badge if there is one available. If not, you should wear FAMOA as you are an FA official and that is why you're appointed to that game, not County.
For ALL County games, it is a requirement to wear the badge of the County, no exceptions/excuses.
Other than that, wear what you want. Personally, I always wore FAMOA and then added a second badge, depending on league/competition that I was officiating in.