AOPA has a good article on this topic:
- You're required to present pilot and medical certificates (plus logbook if required for flight)
- The inspector may not detain you
- The inspector may not board the aircraft without informing you
But an inspector is authorized to check the following items:
- The airworthiness certificate
- The aircraft registration
- The operating handbook
- The weight and balance information
- The minimum equipment list (if applicable)
- Aeronautical charts (if applicable)
- The general airworthiness of the aircraft
- The ELT battery
- A VOR check
- The seats/safety belts.
In general, saying as little as possible seems to be a good plan:
If the ramp check is due to a possible violation, anything you say or
do may be used against you.
If you have enrolled in AOPA's Legal Services Plan, call AOPA's Pilot
Information Center at 800/USA-AOPA immediately. The consequences for
even minor infractions can be far more serious than you might think.
As for a search warrant, a ramp check is an FAA administrative process, not a law enforcement action. The issue of law enforcement checking private aircraft is a hot topic currently and AOPA provides a checklist in case you are stopped for search. Their recommendation is to refuse the search (unless they have a warrant, of course) but not interfere if they go ahead and do it anyway.