I have personally slept in my car on cross-country road trips and ski trips for years. Only twice was I awakened by a police officer. Both times were early in my car-sleeping career, in poorly chosen parking spots, and in the deep south. I have had no other unpleasant experiences sleeping in the car except for locking my keys in the car after waking up and stepping outside without my key ring.
For me sleeping in the car means an air mattress and subzero sleeping bag in the back with the rear seats folded forward in a Honda Civic. Yes, it is not as comfortable as a hotel, and I will usually find a cheap hotel/public shower to visit every now and then to freshen up, but as a means to save money when traveling, it comes second only to crashing friends' couches (also highly recommended).
My first experiment with sleeping in the car came on a 45 day round-the-US road trip, where I spent about 35 of the 45 days sleeping in the vehicle. Even at the cheapest hotels, I would have spent over $1000.00 on lodging alone for those days - money much better spent on fuel, food, and entertainment/recreation.
My next experiment was car-camping in the parking lot at a ski area (in the designated overnight area). Having found this to be surprisingly comfortable (and convenient), I have gone on to spend the last two years camping inside the car on most of my snowboarding trips, with no problems.
Keys to being safe are choosing a good parking spot, making sure you have sufficient fuel/food/facilities at hand, and being prepared for the conditions. Make sure you have a plan for where to pee in the morning, and you should be fine. Just don't park in residential zones, on dead-end roads, or anywhere that you are an obvious target for police/criminals.
I have had success with highway viewpoints, city streets, and many areas in between. It is no problem to sleep in the middle of the city if you are tired enough, and have a sun-shade to block any street lights. I would recommend an area with access to a restroom if possible, but if none is available, just make sure you have some t.p. or other appropriate wiping device at hand.
Feel free to touch base with me if you have questions about the ins and outs, or anything specific to sleeping in the car in snow zones. Remember that snow plows are big and heavy, and you are not - plan appropriately.
Happy trails!