Within Lonely Planet's Moscow Guide, there was a section called "The Great Moscow Police Tourist Rip-off." Mention of this scam didn't appear within the Rough Guide to Moscow, and I didn't see it take place for the first 2 days I spent wandering around the city. I assumed it was a form of corruption that had been purged from police practice... that ism until I saw it in action at Krasnopresnenskaya metro station.
Moscow's metro system is known for being safer than its St. Petersburg counterpart. There is a stronger police presence within the Moscow stations, which decreases the ability for criminals to engage in pick-pocketing and other petty crimes. Unfortunately, one officer working at the top of the escalator leading down to the brown "ring" line was supplementing his income by harassing tourists.
He stopped everyone that could be identified as non-Russian to check their travel documents. He was so adept at his side job that he could round up numerous tourists at once rather than just concentrating on them one at a time. My nephew and I were fortunate enough to notice the shakedown and avoid it.
From what I've read, officers like this will review your travel documents (passport and visa) and find an error, even if nothing is actually wrong. They will present you with the option of visiting the local police station and paying a fine to correct the problem, which could take hours, or letting you pay them directly. The fine can be as much as 2,000 rubles ($74), and the officer may even provide you with a phony receipt.
To avoid being scammed by these unscrupulous types, my tour director recommended that you act as stupid as possible. For tourists staying in local accommodations, it is common to leave your passport and visa with the hotel until the end of your stay. The police are very aware of this practice, so you should state that you are a tourist and show them your hotel keycard. If the police insist that you are in violation, ask them to write out the ticket for you. Here, you have a friend, and it is called paperwork. Remember, these officers are trying to take advantage of a quick money-making enterprise. They don't want to spend the time and effort to complete a bunch of forms when they can find a more compliant victim.
If you find yourself in this situation, Lonely Planet suggests that you make a note of the officer's seven-digit ID number before handing over your documents. Apparently, personal accountability can be threatening. Additionally, having a mobile phone handy to state you are going to notify your country's embassy of the problem is a strong deterrent.
If you ensure that you don't look and act the part of a tourist, I think you will avoid the problem all together. However, use these tips if you aren't so lucky and be aware of the officer awaiting your arrival at Krasnopresnenskaya metro station