The short answer is that any single shot action OTHER THAN a bolt action is legal for general use in ASSRA events...some specify certain restrictions, but this is not a general thing.  Any falling block action (since you mention them specifically) should fit the bill nicely.  The OTHER criterion is that the bullet must be lead alloy only without jacket or metallic gas check...what we refer to as a "plain base lead bullet."

As for the .22 Hornet, I will have to defer to others on that as I have never shot that caliber much and never attempted to load it with the plain base lead bullets which are required for ASSRA competition.  The smallest centerfires that are generally seen on the line are the "quarterbores" (.25 cals) including the .25 Hornet that some folks like our xxgrampa seem to like pretty well.  HTH and I am sure others will chime in on this as well.

Froggie

Falling blocks, tipping blocks, sliding blocks, rolling blocks,pivoting actions, trapdoors, break-opens, and probably even more, as was said by the Frog, anything but the bolt action goes OK.

  Perhaps through your local library and its interlibrary loan service you can get some of the late Frank DeHasse's books on single shot actions.  There are several. I don;t know their titles, maybe somebody else can chime in here and list them.  they would probably give you as complete a listing along with mechanical descriptions and their pros-n-cons as anyone can find.  I have a recollection that his family is still selling them but I may be wrong about that.

I suspect that some of the reason you dont see too many of the calibres like the 22 hornet is that the lighter weight bullets are more subject to wind drift, the use of cast plain-based-bullets-only holds velocity down, and smaller cast bullets are just a whole lot more finicky about loads.  Most of the ASSRA shooters use some sort of Breech seating system (see the FAQ on the main page and other introductory material) andit is a whole lot easier to physically manipulate larger cases and bullets through the process. 

Maybe its also just because no one has really given it a real serious try.  After all there may be a bit of a unconscious residual knee-reflex reaction among dedicated Schuetzen "traditionalists" (perhaps an oxymoron) against varmint cartrdges due to the loudly lamented history of converting single shots into varmint rifles in previous generations.

There is no reason a 22 hornet could not work, provided that someone was willing to put the labor into it, knowing that the light bullets might restrict accuracy in anything other than dead calm conditions--or it'll help you learn wind reading skills quickly.

  If thats what you have, or thats what you want to explore, bring it and come shooting with us.  Most of us if we are honest are competing against ourselves anyway and a personal best, no matter what you are shooting, is a good feeling.  We shoot 22 rimfires at 200 yards in our regular matches and at 100 yrds in the 22 only matches (at Etna Green that is--other clubs have their own arrangements) so we know a slow 22 bullet, even with fixed ammo, can be shot well

P.S.    Also I was looking for specific actions, like a Hock or a Peregrine or a Clerke.
Thanks again
Slick4k

If all I had was a Hornet, I'd give it a try, albeit with a heavier than standard-weight bullet (60 gr.+) and possibly of a harder alloy. Shootin something is better thtan not shootin!