Registration has a quite different function to Visas and Passports
In RUS, a Passport is your Identity Document, that should be with you at all times.
A Visa, whether stuck in the Passport or a separate document (such as my 'Exit' visa) shows your entitlement to be in RUS at that date - or not!
Registration is a hangover from the soviet days of watching and controlling people and their movements. It is essentially a means of knowing your address - or recent address. It is required for functions such as opening bank accounts, obtaining credit, in support of other government burocracy like obtaining Residency, Driving Licences etc. But the Law remains, even though largely ignored by Russians and is gradually being relaxed for foreigners (like recently extending the 5 day rule to 7 working days).
When stopped by police for personal documents they are interested in your identity and right to be in RUS. In border areas and sensitive parts like Dagestan, Chechnia etc, such checks are, for security reasons, far more frequent and thorough. They are not interested in seeing your Migration Card or Registration. But if you proffer them they will realise you don't understand so will look at them and try to find something wrong with which to fund their evening meal.
A frequent difficulty for visitors is not knowing what the different uniforms represent, so not knowing who has stopped you, why and possibly not understanding the language sufficiently to know exactly what is being asking for. The normal stern or unhappy looking facial set of most Russians does not help either. The natural response on being uncertain is to offer everything you have - sooner that just the specific thing being asked for.
You can be stopped anywhere by Roads Police, Militsia (town) Police, Border Police, National Security Police, Customs and Military. Each has a different interest. None is interested in Registration these days.
Roads Police (ДПС) have no right to see anything other than your vehicle and driving documents. But they will try it on.
The old Tourist Police that terrorised people for fines/bribes in Red Square and the Kremlin was disbanded as it served no useful purpose and was having an adverse effect on the image of Russia gained by tourists.
The rule is everyone should have Registration within 7 working days (weekends and public holidays do not count) of arrival - or the last Registration. Even Russians should, by Law, get a new registration when away from home for those periods - but no one does!
All that said, if you get into some trouble they will naturally look at everything to add to the heap of trouble against you. So get a Registration where you can and keep proof of travelling (petrol receipts etc) if ever challenged about no current registration.