That describes the conditions around here a lot of the time. My current shoes, Karrimor, are terrible on wet concrete slabs. On ice they have no grip. My time is really limited so if it is cold for a week I just don't get any running done that week.
I've researched. Shoes with studs seem over the top and sound like they'd skate and slip on wet concrete.
Saucony shoes with Arctic Grip sound like a better solution but none of the shops have them in.
What else is there?
I did ask in shops but they were not very helpful (suggesting getting in a car and driving to a place where I could run offroad - seriously - I run because it is convenient).