Can you run safely in the Snow? Yes, but...
NERVOUS? Then don't risk it. Pretty simple but it needs to be said.
Fresh snow is usually pretty safe to run on, but take it easy, especially the first few minutes. This will allow you to get a feel for under-foot conditions and build your confidence. Take smaller steps and look out for patches of ice under the snow.
If you plan to hit trails...
Because trails haven't been scooped, you are more likely to find snow than ice in most places, which means better traction and less chance of slipping. Running in fresh snow means a little more work, since it's softer. Treat that like running on the beach and keep your run shorter, knowing it will engage new muscles. Also it's not to time to lengthen your stride... keep them short and work on your cadence.
What to wear to run in the snow?
Depending on the day you might find long sleeves and tights are plenty! That’s what makes New England winters so crazy. Other days it feels like full on Artic Winds, in which case:
Trail running shoes
Winter running gloves
Winter running tights (Fleece lined is awesome)
Winter running jacket
The second part of running on snow, is being prepared for the idea that you might find some ice.
Get Winterized Just like when Rocky's babysitters needed chains on their tires in Russian for the mountains (yeah Rocky IV reference), you need to winterize those shoes.
If you aren't going to lace up those trail runners, adding a spike or other gripping tool to your shoes is a great way to enhance traction.
There are a lot of options depending on your budget:
put screws in to your existing shoes Length of the screw depends on the sole...please use the short tiny ones
buy a pullover for your shoes that has spikes (like a yak trak)
invest in quality shoes if you’ll be doing a lot of ice (Trail rated shoes with deep lugs)