Where can the requirements for US aircraft registration be found?
The registration procedure is outlined in FAA Form AFS-750-94, Information to Aid in the Registration of U.S. Civil Aircraft. See also the full list of FAA registration forms.
3. The owner of an aircraft must apply to the Aircraft Registration Branch, P.O. Box 25504, Oklahoma City, Oklahoma 73125-0504, to obtain a certificate of registration. The applicant must submit an Aircraft Registration Application, AC Form 8050-1, accompanied by evidence of ownership and a $5 registration fee.
Requirements for registration of aircraft are contained in the Federal Aviation Regulations, Part 47, “Aircraft Registration.” (For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402.)
Basically, the registrant (whether an individual or a corporation, trustee, etc., all of which are defined in Part 47) must be able to prove legal ownership of the airplane.
What is the penalty for flying an aircraft without a valid registration?
The violator would be subject to any of the standard FAA enforcement actions: administrative action (a "Warning Notice" or "Letter of Correction" which goes on your record), reexamination ("709 ride"), certification action (suspension or revocation), or civil/criminal penalties. The exact penalty would likely depend on the circumstances: whether the violation was intentional and the degree of the pilot's negligence under the FARs.
What should someone do if they believe an aircraft to be operating without valid registration, and how could they determine this?
The FAA's website recommends reporting FAR violations by calling the FAA's toll-free telephone number, 1-866-TELL-FAA.