The problem with the 597 is the magazines. If you have your dual bolt guide springs set correctly AND you get a good magazine it should be pretty reliable. But so many of the factory magazines are duds. The manufacturing tolerances on them are a joke. The 10/22 on the other hand has magazines that are so reliable it's almost a joke too. Except the jokes not on YOU!
Yes the standard 10/22's will handle Stingers just fine. I've used them with no problems at all. I know it's already been mentioned but the Stingers do have a longer case and it's not recommended to use them in a match barrel. I thought it had something to do with the longer brass damaging the chamber because a match barrel would have tighter tolerances in the chamber? Something like that anyway. It's not that you can't shoot them it may shoot them just fine but you could be damaging your target barrel. That's what I took from it anyway.
Anyway, I agree with everyone about Stingers not being that great. I've seen some tests at longer range where the minimags had more penetration! They may not be "hyper velocity" but they are 40gr. I just buy the CCI AR Tactical 22LR's myself. They are very similar in spec to the minimag (40gr copper platted 1,200 FPS) and they sell them in a box of 375 so it works out a little cheaper than the minimags. The Velocitors are also 40gr and just over 1,400 FPS. But once again it's expensive buying those 50 packs!