After a little searching I have not seen any specifics. Did we bury the remnants somewhere? Are they in storage somewhere? How much were they able to "put back together" in analyzing the accidents?
The fragments of both shuttles are in storage. Challenger's remains, totaling about 30% of the shuttle and 75% of the crew area, are stored in a pair of retired missile silos at Cape Canaveral.
Something like 84000 pieces of debris from Columbia were recovered from all across the American Southwest, and after cataloguing and study into what caused the accident, were stored in a 7000sq ft room at the Kennedy Space Center's Vehicle Assembly Building. Researchers can request to visit the space, but it's not open to the public in any way.
In mid-2015, NASA opened a permanent exhibit entitled "Forever Remembered" at the KSC. There, the public can see pieces of the two shuttles and tributes to the crewmembers.