DougK.....I assume you mean single shot break action or falling block.
However if you want out of the box accuracy single shot, for less then $1000 then a single shot rifle made by Cooper Arms of Montana is your Gun. I have a friend, who works at gun shop, talked the owner into being a Cooper dealer, you have to buy 4 or 5 a year, anyways he owns five (17hmr,22br,222rem,6mmppc and 6.5x284) not one of these gun shoot's over .5 and most shoot a lot less the 6.5x284 is shooting in .2's. He has sold 22lr/22mag's repeaters and none have shot groups over .5inches at 100 meter. If you want the most accurate single shot rifle for the under $1000 a base Cooper bolt action is your gun, MSRP is above $1000 but, out the door price on the base gun will be less then grand, not by much mind you. There maybe onezy two-zy out there Ruger or H&R that
shoot's less then .5 inch but, on average they don't. I own two Ruger #1 a B and V model a 6mm made in 1975 that shoot's .75 ,but has been worked over and the heavy 22-250 that shoot's in the .6's Yes, there are Number one's that shoot less then .5 but, they're few and far between, most Ruger #1 owners are very happy if they shoot in the .7's H&R's have a strong presence here and for grand you could buy and sell them until you got yourself a tack driver and have cash left over.
My friend has sold 14 Cooper's or 12 with 2 on order, something like that, anyway's not one hasn't been a what I call a real shooter.