What is the good choice for the running shoes for winter conditions in city (mostly running on asphalt)? When the snow is shoveld, normal running shoes would be no problem, but when the snow falling, they are soaking immediately and they have poor adhesion.
So, I was thinking about running in light trekking shoes with elastic soles, because of good adhesion and light membrane. But, aren't such shoes too overloading for ankles?
For snowier conditions, it is common in the ultra community to take an old pair of shoes and screw in a number of metal hex screws into the sole from the bottom leaving enough of the screw proud to stick into the snow.
I've never had to try it myself but I'm reliably informed it works a treat.
As stated by Graham in his comment, I would recommend using ice traction device like this one or this one. It will provide you with the missing grip in winter.
You should definitely keep using your running shoes because they are still better suited for running even in winter conditions.
What sort of snow conditions are you running in? For dry, powdery snow, the best option is a pair running shoes that have aggressive tread (search for "trail running shoes"), but in wet, icy snow, metal screws or spikes will give you the extra grip you're looking for. I can't think of anything that will help more than it will hurt on icy pavement other than proceeding with caution.
You should be aware that traction devices have their own limitations. Metal spikes or screws will severely reduce the amount of traction you get on cleared pavement. On bare pavement or hard ice, the metal screw technique Sarge mentioned also has the potential to cause blisters and irritation just like using XC spikes on pavement does. If you need the traction devices for only a short portion of your run, you will be better off just being careful on the slippery parts. That seems to be the approach a vast majority of competitive runners take, even in severe winters in Minnesota.
Icebug specializes in shoes made for running on ice or other very slippery surfaces.
Except for the type of shoes, I wouldn't use shoes with a thin sole, because when the road is covered with snow it's very easy to miss some dirt or other small objects where you place your shoes on.
I even would say, it is better to run slightly more on your mid-foot and not heel, to prevent twisting your ankle.
