Studded snow tires for bicycles exist.
They last years if you put them on when it starts getting snowy and take them off in the spring.
Get the ones with proper carbide studs - they're more expensive than steel studs, but last a lot longer (4-5 winters is doable with daily riding, but beyond that you lose too many studs to really have good traction). They're really luggy too for tossing snow out.
They're scary on dry pavement if you're used to good rubber - I described them as "60% of dry pavement grip on dry pavement, 50% of dry pavement grip on a sheet of ice." Do NOT push it hard through corners on them, or you'll get off the stud track and are back to hard rubber on ice, which has no traction.
But, you can ride through basically anything with them. You can do stoppies on a sheet of ice with them. And you can be running down a road, hearing the purr of the studs on the pavement, and when it all goes silent, realize you hit black ice you didn't even see - and are gripping just fine.
They are absolutely worth the cost (I think a good pair should run you $100-$150 installed) if you're biking in the winter. The looks you get are totally worth it. :)
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I don't bike as much anymore (work from home, live on a rural 55mph road a dozen miles out of town), but I ride my sidecar motorcycle all winter long (2WD - you can lock the sidecar wheel in for low traction conditions). I share the same joy in getting, "You... rode in this???" comments/looks. :) And when I biked everywhere, I definitely got some good comments in the winter. It's fun!
You don't even have to do it that much to get the reputation as a winter riding badass. Just a few times is enough!