I left my chains in my trunk for a year once. It was mostly because I didn't bother to take them out, but it sure would have been handy if there was a freak ice storm.
The other issue is that studs aren't really useful on snow. The work on ice by digging in and breaking it up.
There are some equivalents for shoes. I've tried walking on ice and it's pretty bad. I've bought these things called Yak Trax, which are basically coils similar to the coils on some tire cables. They didn't seem to work too well on a carpet (never actually used them outdoors), but I think they're designed to work better when roughed up. The current version seems a little different than what I bought. I heard it worked better on deep snow than on ice.
The other alternative is mini-crampons. There are a whole host of different types, including the equivalent of screws in a base. I've even heard of some people just tapping screws into their shoes if there's enough tread there, but it seems like there's a risk of them punching into the foot. And finally there are some that use tungsten carbide studs. I tried those once. They weren't quite like tire studs that eventually wear flush with the tread. But on an icy sidewalk I stayed firmly planted as long as I kept my foot flat.