I would follow the path of the masters regarding what action to use. Harry Pope, AW Peterson, CC Johnson, Neidner, Hubalek, and Eric Johnson built some of the most accurate target single shots ever and favored the Ballard action. Eric Johnson developed probably the best extractor system and Pope's lightened, speed-locked, self-cocking hammer makes the Ballards hard to beat (though Pope also used the same system on High Walls). Ballards can be tuned to breech up as tight as you could want, 'will push rounds into a tight chamber, their double-set triggers are silky smooth, and the through bolt will ensure the butt is solid to the action. 
A lot of aspects of building a gun have been mentioned (barrels, actiions, etc), you shouldn't overlook the importance of proper extractor fit to prevent buldged brass. At the .22 BPCR silhouette match at Raton last year I picked up some empty cases and was shocked at how many were buldged at the rim. The year before last a shooter was nearly blinded when a case exploded. Luckily she had her glasses on. I've seen a lot of buldged brass coming out of Winchesters with kicker extractors. I've been told the cases apparently aren't well supported in the chamber because of the large extractor. Because of the small size of the chambers, extra care must be taken to make everything just right. This is much less of a problem with big calibers, again info from a very reputable gunsmith.

Man,  I can't believe I'm going to say this........you start out with a ruger 10/22 magazine.  Buy a force action,buy a shilen barrel(threaded), go through the brownells book and buy everything you can titanium, oversized pins and bushings,a bell and carlson bench stock(fully adj.).  First crown the barrel at 11 degrees,  chamber it to a match chamber, bed the action in the stock, go to your local sporting goods shop and buy the best scope you can afford(this one happended to be a 6-24 swarovski), buy every style of 22lr ammo the store has in stock, a bundle of targets and off to the range.  I just so happen to find cci green tag and std shot the best.  Its amazing what a few thousand well spent dollars can doo for your accuracy......I didn't just post that on a single shot page did I??? sorry to offend anybody.  Just couldn't resist!!!!


kinglooie

Sure.. you can pour money into a 10/22 and make it shoot.. (been there, done that, so to speak).
It'll actually be CHEAPER to build a single shot, or bolt action for that matter, that will out-shoot a 10/22. Even a fully rebuild "race gun".

I've had guys tell me their 10/22's will outshoot my Anschutz prone rifle.
None of them have proved it (or even showed up to try...). 

Back to the question at hand;
I'd pick the Stevens 44, cause I have one and I'm thinking of rebuilding that into a nice offhand .22. 

Paul F.

I thought you might go with the trapdoor Dave.  Ballards are nice though.  Not too many .22 trapdoors around. Kind of unique. Scarce.


                                                                      Joe.