For English, a running braid should be perfectly fine in any hunter/dressage type class. Western, horses aren't braided. They do band the manes for stock types, but for a local show, I'm not sure if I would unless you find everyone else is also.
As far as the ribbon browbands, I've never actually seen one at a show. They don't seem to be a "thing" here in the US except in a few select circuits.