I work at Lane Motor Museum in Nashville, TN. We have used, almost exclusively, Thomas Sunday Transport since opening in 2003. I'm sure we've had close to 100 cars delivered with them, and never a problem. The owner/drivers are conscientious, punctual, and communicative. We deal with a lot of other haulers as well when institutions or individuals arrange their own transport - any transport is only as good as the driver, although decent equipment counts for a lot as well.
We ship and recieve everything from 6-figure classics to non-running and sometimes non-rolling oddball three wheelers, and Thomas Sunday never misses a beat. Lots of comparable carriers, like Intercity, Passport, and Reliable, but I don't have much direct experience with any of them but Reliable.
On another list I frequent (BMW 2002 FAQ) there are a lot of folks who swear by DAS Auto for transport, but again I have no experience with them personally, only anecdotally.
David