Random police checkes are, apparently, not a Moscow-only phenomenon. Yesterday I spoke to a chap, considerably more respectable that me in appearance and manner, who nevertheless is stopped by the police with uncomfortable regularity. This winter that happened twice. I could not get all the details from him because it was, he said, “too unpleasant to talk about”.
It appears that some are checked a lot, eg. once a month, while others, like myself, for whatever reason, may run around doing their evil deeds for a year or two without attracting the attention of authorities. I reaffirm my advice to carry your passport and registration on you. In our out of Moscow. If papers are in order and you are sober you will be let go with 99% probability. At least have copies, if possible notarized (there are conflicting reports about it being possible to put public notary’s stamp on passport copies) or with some official seal on them. And have a few telephone numbers, uncluding that of your embassy if you are a citizen of one of these countires that give a damn about their people abroad (bad reports are heared about the US embassies), on you. No, you are not likely to be beaten unless, but losing your money and valuables is realistic and probably, and unlike ordinary honest thieves and robbers, Russian cops will go out of their way to make your experience memorable.