Question

what sort of covering would be put on the walls of a sound-proof studio

Answer 1

We need to put a covering that does not reflect the sound.  We need to put a covering  that absorbs sound energy.  The sound waves should hit the wall and the reflected waves should get destructively combined so that
      the sound vibrations do not pass through the walls.
      the resound does not affect the proceedings in side the studio.

So a soft coating may be used.
The surface will be rough, with small projections coming out of the surface and the surface is uneven.  The reflections will be scattered in all directions.
Some wall paper type, or cloth covering works good.