I'm not an expat, but I whant to advise you to read one interesting magazine - "Unbound", that posts true stories by teachers working for BKC IH Moscow. For example, there is an article by Keny Kurata telling about his experience of being stopped by Moscow police.

Keny writes: "One of the most important bit of advice I can probably give is how to avoid being asked for your documents. The best way to avoid the police is to look busy. If you are in a rush to get somewhere, the police are not likely to stop you (unless of course you are doing something suspicious). Most of the time, this is not a problem since we are always in a rush to get somewhere anyway. The great majority of the times I have been asked for my documents were when I had nothing to do or I was waiting for a train, a bus or a friend. "

See the whole article here http://unboundmagazine.wordpress.com/20 ... se-part-1/