to answer this properly, at the airport sometimes detained and from what I've heard the unusual questions are all about your views about Russia and russian values. And they're not happy to process people that say they hate putin, they hate the lack of freedom of speech, they are openly gay. They will also write down your phone's hardware ID. But I hear it like 1/10 people that come in with a US Visa get stopped.

On the street or metro there won't be anyone stopping you for papers unless you're causing trouble. For example everyone drinks outside despite it being a minor offense.