I wear polarized sunglasses on the bike, and it sounds like the OP has as well until now, but my thoughts:
The frame/style is really up to you - whatever fits your face, and under your helmet comfortably, is key. Any of the name brands, if purchased from a reputable dealer, and having a sturdy design, should give years of service. Suggest bringing your helmet when trying them on. If it fits comfortably, and you like the look, you're probably good to go.
Mine are prescription, so they're just stupidly $ (be very thankful if you don't wear glasses!!!) but you get the option to customize in terms of lens color/coatings/filters etc. so I've had the opportunity to try a few different options and evaluate them on their own independently of one another.
In terms of polarization: in addition to making funny colors when used under some helmet visors, they often make things that are black (like tar strips) oddly iridescent. As others have mentioned, LCD displays may or may not be visible, depending on their orientation compared to your head.
At the same time, it's said that polarized sunglasses could mask dark oil puddles etc. on the road surface, making them "less safe" for motorcyclists by supposedly cutting the glare that we use to identify them. FWIW.
At the end of the day, I still use polarized sunglasses for riding because I also use them off the bike, and just ignore the psychedelic colors when riding, remaining aware that they could potentially mask oil slicks. Any close-fitting "sport" style "wrap-around" frames probably have the best odds of fitting under the helmet (from whatever you favorite brand is). But everyone's head and helmet are different; all you can do is try them on.
I've only had lousy luck with progressive tinted lenses (the UV filter on my helmet visor [and truck windshield] prevents the progressive tinting from being activated).