Not sure why Solve fails but Reduce seems to work:
Solve[x^(3/2) == -1, x, Method -> Reduce]
(* {{x -> -(-1)^(1/3)}, {x -> (-1)^(2/3)}} *)
(As you may know, x^(3/2) is the principal root, and so we don't get all solutions to the rationalized equation, (x^3) == (-1)^2.)
