Were there specific FAA (or other relevant regulatory body) pilot licensing or certifications required to fly the Space Shuttle? Are they specific to the Space Shuttle, or would it have fallen under an existing pilot license scheme?
FAA considers space shuttle either a military aircraft or spacecraft and hence its rules don't apply on space shuttle.
However, requirements to be a pilot for a space shuttle are:
... at least 1,000 hours pilot-in-command time on jet aircraft
It means that you need type rating to operate a heavy jet aircraft (not light jet). Weight of an empty space shuttle is 172,000 lb (78,000 kg)1. In addition, you need the additional requirements mentioned by NASA (link above), which include education and health standards to meet.
Operating empty weight of Airbus A380 is 610,000 lbs.
@Farhan covers the NASA requirements but legally, According to the FAA FAR's
(b) Each member of a flight crew must demonstrate an ability to withstand the stresses of space flight, which may include high acceleration or deceleration, microgravity, and vibration, in sufficient condition to safely carry out his or her duties so that the vehicle will not harm the public.
(c) A pilot and a remote operator must—
(1) Possess and carry an FAA pilot certificate with an instrument rating.
The effective minimum rating would be a pilot with an instrument rating as this would be required to operate the shuttle training aircraft, which they would do at least 1,000 times and often from Class A airspace at 37,000 feet.