Hulls are registered with the Interstellar Ship Authority, and as part of that registration, they are assigned a registry number, and are inspected to make sure they're in compliance with ISA regulations. That registry number can be used to look up the ship owner, history, and if the ship has been reported stolen.
While the ISA isn't a law enforcement organization (it actively works to stay out of law enforcement), it does have it's own authority and can flag or revoke a registry, with the hopes local law enforcement will assist them in enforcment. Typically organizations like this have very little teeth, but the ISA is quite the opposite.
An oft repeated phrase is, "The ISA doesn't have many rules, but the ones they do are absolute."
The following are requirements to register for a registry number:
Once a registration is complete and a registry number assigned, there is a follow up inspection within 6 terran weeks to ensure:
There is a 'military exception' to registration for all major governments that automatically allows registration of all military vessels as long as there is not overlap with an existing registry codes.
In times of war, flying without a registry, or with a placeholder was very common in the past but is less so now.
Registry numbers are in the following format:
<Prefix: 2-5 letters> - <Identification: 1 to 9 numeric digits> <Differentiator: 1-3 letters (optional)>
Examples: AR-1, SD-13, XNIL-007, RVN-1849, NCC-1701 D
Typically, the letter designation is either a manufacturer code or model abbreviation, with the numbers working more like a serial number. Most owners don't bother getting a custom registry unless they want the vanity aspect of it.
For an extra 50 credit fee, you can specify your own custom registry number, as long as it complies with the rules above. If you want one that doesn't comply (OUTATIME being one of the most common requested), it's possible but subject to approval by the ISA board of directors. It can take years to get approved and the number of exceptions are very rare because of this.