Hello and welcome to OT, barryhanna!
I have a Browning BL22 as well, and really enjoy shooting that little rifle. Mine isn't very accurate and the trigger pull is heavy and rather rough, but it is a sweet handling, well-balanced, light, well-made, dependable, and nice looking little rifle!
In my personal opinion, that rifle is at its best without a scope, just using the iron sights. I had a scope on it for awhile, and to me, it detracted from the very thing that makes that gun fun to shoot -- it's slender, unobtrusive, and easy handling -- so I took it off. On top of that, I couldn't shoot any more accurately at typical rimfire distances with the scope as I could without, since extreme accuracy isn't this rifle's strong point. It also has a very short receiver, and therefore, all but very short scopes looks odd on the rifle to me, not to mention that even a moderate weight scope makes the rifle top heavy due to its very light weight.
If I were scoping this rifle, I would choose a fixed 4X rimfire model scope, or at least a low powered variable. Any of these would be a good choice for this rifle in my opinion (the Zeiss being the best of the bunch and the Weavers the most affordable if you have a tight budget):
http://www.swfa.com/pc-6697-259-zeiss-4x32-conquest-rifle-scope.aspx - http://www.swfa.com/pc-6697-259-zeiss-4x32-conquest-rifle-scope.aspx
http://www.swfa.com/pc-3621-100-leupold-2-7x28-vx-i-rimfire-riflescope.aspx - http://www.swfa.com/pc-3621-100-leupold-2-7x28-vx-i-rimfire-riflescope.aspx
http://www.swfa.com/pc-3619-100-leupold-4x28-fx-i-rimfire-riflescope.aspx - http://www.swfa.com/pc-3619-100-leupold-4x28-fx-i-rimfire-riflescope.aspx
http://www.swfa.com/pc-10607-1514-new-sightron-4x32-sii-riflescope.aspx - http://www.swfa.com/pc-10607-1514-new-sightron-4x32-sii-riflescope.aspx
http://www.swfa.com/pc-2344-257-weaver-25-7x28-classic-rimfire-rifle-scope.aspx - http://www.swfa.com/pc-2344-257-weaver-25-7x28-classic-rimfire-rifle-scope.aspx
http://www.swfa.com/pc-2343-257-weaver-4x28-classic-rimfire-rifle-scope.aspx - http://www.swfa.com/pc-2343-257-weaver-4x28-classic-rimfire-rifle-scope.aspx
I believe Sightron also makes a variable scope in their S1 series intended for rimfire use, but I don't know the specific model off-hand. I don't own any Sightron scopes, but those that do say they are good scopes. Whatever you do, be sure to choose a scope that is parallax adjusted to 50 yards. Most scopes you buy are intended for centerfire rifles and are therefore parallax adjusted for 100 yds/meters, which makes them not so great for rimfire use, because you will experience extreme parallax (reticle not focused on the same plane as the target image) problems that hurt your accuracy potential at short range. Unfortunately, there aren't very many "good" rimfire scopes available anymore. Most of the dedicated rimfire scopes are just cheap scopes. I also would highly encourage you not to select a very large, high magnification scope, as it is totally unnecessary on that rifle.
------------- Ted
Money can't buy happiness... but it's much more comfortable to cry in a Porsche than on a bicycle.
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