Everywhere I look, there is snow or ice. It snowed. It melted. It froze. Every sidewalk, every road, everywhere you look, there is something. It's difficult to walk, much less run. And, if it's not ice or snow, it's slush.
The risk of slipping and falling is very, very real. The risk of getting sick because your feet are wet is very, very real.
So how can you make running in these conditions safer?
First, look at how you are running. When running on slippery surfaces you want to be careful not to over stride. This puts your center of mass too far ahead of you and can increase the chances of you losing your balance. Try to have your foot land directly beneath your hips, this will afford you more stability and more of your foot will make contact with the ground, providing a greater surface area to land on. Also, just like when you are driving, you don't want to brake quickly on ice. When you strike the ground with your heel it slows you down, effectively acting like a brake in your car.
Snow and ice bring salt, which can destroy shoes. You may be tempted to reach for a beat-up old pair that you don't care about. Don't. You want shoes with good traction and enough foam to cushion any impact. Ruining a pair of shoes is better than breaking your ankle. Look at shoes made specifically for running in cold weather, such as the adidas Energy Boost 2 ATR running shoes.
However, your best bet may be trail shoes. You want shoes that will keep your feet warm and dry and that offer more traction. The Salomon X-Scream 3D is a good bet, as is the Salomon X-Mission 2, the ASICS GT-2000 3 and the Hoka One One Challenger ATR.
Avoid racing flats or any shoes which have little to no tread. Of course, in really icy conditions, you can always add something like a Yaktrax.
Obviously, if you can run on surfaces which have been cleared, you will be much better off. If that isn't an option, your next best bet is running on snow that is already packed down, but please be careful. There may be things underneath the snow that you cannot see.
If you find yourself in an extremely icy area please put your safety first, slow down, and walk carefully. You may not want to miss your run, but if you fall and hurt yourself, you will end up missing much more than just one. Please, be careful out there.