Denny has the right idea. You don't need the best but you need something better than the junk most makers sell as a 22 scope. Then an AO is an absolute must if you want to shoot groups of little bitty holes.
The Simmons 22 Mag with the AO is a good deal at about $60. The Muellers are good for the $. I have 2 APVs. They are around $120.
Check out Rimfirecentral.com for some other input.
The old Winchester has this little 1/2" diameter scope, probably as old as it is. There was an old Weaver 4x in the gun cabinet that came off Dad's old 300 savage, I added it to the Rem 22 pump...nice and clear w/ 1" objectives and very skinny cross hairs, works well for me.
Sorry Moosehunter i took off 3x9 Redfield Widefield and put on Bushnell XLT Trophy 4x12x40 on my ss Marlin 22 mag. I keep my eye sharp with my 870 Rifled bbl. Hopein to get H&R single shot soon and scope that also....later
I have on a ruger 10/22 stainless model wood stock a nice Simmons 3x9x32MM made in japan,the bell has the red ring around it nice set up.I purchased the gun about 2 years ago in a gun shop on the used rack without the scope 150.00 out the door i wold say 99 percent mint made in 1993