In this paper, we examine the computational complexity of enumeration in certain genome rearrangement models. We first show that the Pairwise Rearrangement problem in the Single Cut-and-Join model is sharp-P-complete under polynomial-time Turing reductions. Next, we show that in the Single Cut or Join model, the problem of enumerating all medians is logspace-computable, improving upon the previous polynomial-time bound of Miklós & Smith.