Over here anything pony club you MUST have an approved helmet on, even if its just walking around on the lead rein you must have one.
Anything over jumps in any competition you must have a helmet approved by the show organisers.
Dressage you must wear an approved helmet with propper straps untill.. level 5 and even then you must wear a helmet for all warming up untill the final warm up then you can swap to a top hat or bowler.
And for A&P's on the flat you must wear an approved helmet untill the age of.. I think 18 but it may be 16.