How are aircraft identified as unregistered? When someone operates an unregistered vehicle on public roads we know how they are caught: Police routinely pull people over, and they always check registration of vehicles.  In fact, police will frequently "run tags" without even stopping cars.  They have a realtime connection to a nationwide registry that can report the details of the tagged car and whether a tag is expired or stolen.

Apparently there are a good number of fines and prosecutions for operation of unregistered aircraft.  And at least some are reported by ATC.  I'm going to assume that nobody is foolish enough to file a flight plan listing an unregistered aircraft ID.

So how does ATC check aircraft registration?  Do they have live access to the registry?  When someone calls an ID over the radio are there controllers that routinely check the ID's registration?  Is it illegal to make up an ID for radio interaction with ATC?  Are there people that visually watch for tail numbers and discrepancies with radio calls?
Aircraft don't have to be actually flying for people to notice something fishy going on. In particular, there are always people hanging about at airports keeping an eye on everything going on. Furthermore (at least in the US), aircraft registration details are a matter of public record and can be easily looked up online.
For example, here is the registration detail page for John Travolta's private Boeing 707: https://registry.faa.gov/AircraftInquiry/Search/NNumberResult?nNumberTxt=N707JT
As you can see it contains complete registration details including type, validity status, expiration dates, owner name and address, and airworthiness status.