Here we have a definitive list of safety standards for use in UK competitions.

British Dressage

British – All PAS 015, VG1 - provided they are BSI Kitemarked  
European – VG1 - provided they are BSI Kitemarked 
American – SEI ASTM F1163 04a onwards and SNELL E2001
Australian & New Zealand – AS/NZS 3838 2003 onwards.

British/European - EN1384, BSEN1384 may be worn until 31st December 2016 but not thereafter.

British Showjumping

101.3 Hats. A ‘Riding Hat’, designed to provide protection, must be worn, at all times when mounted by anyone, whether or not a Competitor, at a British Showjumping competition. Harnesses must but properly adjusted and fastened at all times when mounted.
When jumping, including warming up, all Competitors must wear ‘Protective Headwear’ i.e. a hat which meets one of the following standards:
 

British – All PAS 015, VG1 - provided they are BSI Kitemarked
European – VG1 - provided they are BSI Kitemarked
American –All SEI ASTM F1163 04a onwards and SNELL E2001
Australian & New Zealand – AS/NZS 3838 2003 onwards.

British Eventing

Protective Headwear, as listed below, must be worn by Competitors at all times when mounted in BE80(T), BE90, BE100 and BE100 Plus classes.

When jumping, including warming up, all Competitors must wear ‘Protective Headwear’ i.e. a hat which meets one of the following standards:

British – All PAS 015, VG1 - provided they are BSI Kitemarked   
European – VG1 - provided they are BSI Kitemarked  
American –All SEI ASTM F1163 04a onwards and SNELL E2001
Australian & New Zealand – AS/NZS 3838 2003 onwards.

‘Protective Headwear’ must have been checked by an official to make sure that it is labelled with one of the above standards and then marked with a visible current BE ‘hat tag’.

British Horse Society

G22.2: Protective Headwear
‘Protective Headwear’ must be worn at all times by anyone, whether or not a Competitor, riding anywhere at any BRC event. Harnesses must be correctly adjusted and fastened at all times. At all BRC Championships and Qualifiers ‘Protective Headwear’ must have been checked by an official to make sure that it is labelled with one of the accepted standards and then marked with a visible BRC ORANGE hat tag. Failure to do so will incur elimination.  Due to a change in the accepted standards for this year, BRC BLUE hat tags will no longer be accepted. ‘Protective Headwear’ constitutes a hat which meets one of the following standards:

British - PAS 015: 1998 or 2011 provided they are BSI Kitemarked
             VG1 01.040: 2014-12 provided they are BSI Kitemarked
European - VG1 01.040: 2014-12 provided they are BSI Kitemarked
American - ASTM F1163: 2004a or 04a onwards provided they are SEI marked
                 SNELL E2001
Australian and New Zealand - AS/NZS 3838: 2006 onwards provided they are SAI global marked

IMPORTANT CHANGE FOR THE CROSS COUNTRY PHASE: Only a “Jockey Skull” complying to the above standards may be worn for any XC phase. This must not have a peak, any peak type extensions or noticeable protuberances above the eyes or to the front and should have an even round or elliptical shape with a smooth or slightly abrasive surface. A removable hat cover with a light flexible peak may be used if required.

Credit to BETA (British Equestrian Trade Association) for the content on this page.