Bruce, first welcome aboard.  As to your question, the choice is very much up to you for all but a very few types of specific matches.  The "old timers" tend to gravitate toward external adjust scopes made by Unertl, Lyman or some of the lesser known manufacturers.  Meanwhile, a lot of the current shooters are moving toward modern target scopes, especially those from Leupold and Weaver.  The old standard for offhand use was 8X, with benchresters going higher.  Nowadays the offhand guys are getting up into the 24-36 X range along with the BR boys.

For myself, I have a 15X Lyman Super TargetSpot on one offhand rifle and a Unertl BV-20 on the other and I have a 20X SuperTargetSpot on my bench gun.  My rimfire still has a little 8X TargetSpot Jr.

HTH, and again, welcome aboard!

Froggie

PS  Since all of the scopes I shoot are long discontinued, the prices are a little hard to judge.  On the other hand, the popular Weaver T-36 is usually available for about $450 or so.

Lots of variables to consider.  what calibre you considering.  45 cal bullet holes are a lot easier to spot than 22 or 25.  Rifle/action types can enter the calculation.  You going to use the rifle scope as your primary viewing tool or you going to use a spotting scope too?  You planning on shooting offhand, bench, or both?

How hung up are you on the-old tyme-look?  The old external adjust scopes, Lyman targetspots, Unertls etc are really cool looking, but they are also collectors items now and can cost a bunch.    some of the newer ones like the Mitchells etc are seen but  I can't even think of $$$$$ numbers that high

 However you can use a modern internal adjust scope with a simple weaver mount and do just fine.  IN that case most any decent varmint/target scope can get you started.  I started a few years ago with a discontinued Tasco . Something like a 12-24 variable from one of the Sportsman's Warehouse/Cheaper Than Dirt -kind of mail order outfits on a browning M-78 45-70.  (And frankly I have a better gun and scope now, but my scores are not all that much better---'course scope and rifle are not al the equation either)  An issue or two back Dale McGee had an article in the ASSRAJournal about using and mounting modern adjustable scopes.   Frankly just getting started, I'd go that route. At least to start with. You can always upgrade as you develop.
 At most of the EtnaGreen shoots,  used scopes show up on the trading table, usually at very reasonable prices.

now maybe some guys who really know what they are talking about will weigh in

I have a good deal of experience with Simmons 44 magnum scopes. The 6.5x20x44mm with target knobs is a dandy choice and the best value on the planet. Natchez Shooters Supply has them for just under $100.00. I have a couple or three of them (shoot them in .22 silhouette year around) and they are completely repeatable. They actually are more reliable than a couple of Leupolds that I have. This scope, in my opinion, is a $300.00 scope for $100.00. We have a number of fellows using them at the local club and ALL are happy with them.

Moving up the dollar scale, the Weaver is the hands down "high end" champ (T-36 at about $389.00 at Natchez Shooters Supply). This is my current choice for my Schuetzen rifles.

Both are easily mounted on most any single shot. See my article in the Journal that DWS mentioned for all of the details.

Dale53

BP,   The hooked butt plate and the palm rest  absolutly aids in a steadier hold offhand. No if's and's or but's.

As for a scope, I was always afraid of a higher magnifacation scope for bench or offhand, but I was convinced differently.

My first medal I shot a 10 shot offhand match with a 30X STS and just told myself to keep them in the rings, and I did follow some good advice from a very good OH shooter and scored a 191 out of 250. Not the greatest score but it got me a second place medal.

I asked a fellow once why he shoots a 36X scope, his anwser,  "Because I don't have a 40X scope"

One thing that may not be generally known is that the Simmons 6.5x20x44mm has extraordinary range for an internal adjustment scope.

My Weaver scope will go from 25 yards to 200 yards without problems. I absolutely agree with Ray that Burris Signature rings are the "real ticket" for handling both windage AND elevation shortcomings of rifle/scope combinations. I recommend them without reservation.

Dale53