If you choose a hard shell then I would recommend that they meet at least ASTM standard. BETA Level 3 is even better, but they are bulkier. For example Tipperary makes hard shell vest that meet ASTM standard and some that don’t. For the ones that don’t you have no idea if they will do any good in a fall. At least the ones that meet ASTM standards or the higher BETA Level 3, you know will offer some protection because they have to test them against something.
For air vests, there is no USofA national standard. The French just published some standards but is only local to them. Snell has also put out equiestrian air vest standards, but that hasn’t been adopted by any country yet. So it is a bit of a wild west out there. When I bought my Point Two about 4 years ago they say they subjected their vests to the same tests as BETA Level 3 and passed. They use to publish the results but not anymore.
Horse Pilot (who uses the InMotion system) says their air vests meet the new French standards.
Good luck in choosing. When I got my air vest no one else was using them, but now in the jumping world it is becoming quite common, especially amongst the older set. 