I presented this on the Optic forum but did not receive any responses so I am trying it here:
"I am waiting for a new rifle to arrive (LH Cooper 22 mag). It has Leupold bases installed at the factory. A few people recommended I use the Burris Signature series rings.
Since I do not have any knowledge about either my question is: Are these Burris rings compatiable with the Leupold bases? What height would work the best with the Leupold 3x9 rimfire scope?

yes, and I believe the low rings will be the best fit. I have medium Burris rings on my 57M and will change them to low rings at some time in the future.

The matte blue rings were a good match for the finish on my rifle.

Thank you Az Pete. Does your rifle carry the 3x9 Leupold?

No, I have a 4x Nikon on mine. However, for a while I worked for a local Sportsman's Warehouse, and mounted a number of scopes daily. The 3x9 Leupold VX series (40mm objective) will mount on most sporter weight rifles using the current Leupold "Redfield type" low rings. Though some specified Medium rings, the Low rings fit well and don't leave a large gap between the objective bell and barrel. If the rifle had open sights on the sporter weight barrel, the scope would not clear the rear sight, but often would fit behind it, depending on how the customer's eye placement. I beleive the 3x9 Leupold rim fire scope has a 33 mm objective so, I'm about 98% sure the low rings will fit.