Look at Nokian tires.
Nokian is a Finnish tire company known for high-quality winter tires. The WR (now WR G2) is a year-round tire that is decent in the dry while still being a genuine winter tire (with the mountain+snowflake symbol that denotes a true snow tire), and is low-rolling resistance to boot which means better MPG. They are becoming increasingly popular, do a search. Nokians are a little hard to find, you may have to buy them mail-order, and they are not cheap. But being able to leave them on year-round, rather than buying an extra set of wheels and tires, is a plus in my book.
I just noticed that a few sizes are available in run-flat.
http://www.tirefactory.net/wrg2.htm
I run Nokians (Hakkapelita CS or CR, I forget which) year-round on my van and they've been very good both summer and winter, including in the snow.
For the wife's car, we bought a set of Bridgestone Blizzaks for the winter - this was before I learned about the Nokians. They work very well in snow, but treadlife is short. We're only going to get one more winter from them, and then I plan to switch her to Nokian WR's for year-round use.
I should add - winter here in Portland is almost all rain at the lower elevations, but as we drive over to Eastern Washington quite often our winter tires do get to show their stuff in snow and ice.
Also - on the wife's car (Prius) the TPMS system doesn't beep when running the winter wheels/tires without monitors, but the alert symbol is constantly illuminated on the dash. Your local Mazda dealer should be able to tell you how your car will act when running monitor-less tires. Finally, in theory you might want narrower tires for the snow, but in practice I think the stock size is fine.