Varies somewhat. Usually on my side, sometimes on my stomach, sometimes on a diagonal. Usually on my right (at least in my own bed - I seem to pick my left depending on what I'm sleeping on/in and what surrounds me). Usually curled and kind of hugging my pillow. I also like to sleep with a blanket that's at least somewhat heavy, even if not for warmth. It just feels hard to sleep without the feeling of the blanket on top of me.
On cold nights, especially in the winter, I really like grabbing the corners of my blankets closest to the pillow, and wrapping them around me like a cocoon. I have to adjust them and pull them in until no cold air is leaking in. It seems like the only way I can sleep when doing this is on my stomach.
Here's something interesting: I tend to wake up at least once during the wee hours, when I haven't got enough rest yet. When I do so, I like to take a thick pillow (or pillow-doll, more specifically) and lie on it in sort of a modified Child's Pose (the Yoga position), facing the foot of the bed. I'll usually have my blankets at least loosely draped on me. If I wake up again, I might switch back to laying on my side as I normally start out.