There is a polynomial time algorithm by Leighton and Miller for deciding whether cospectral graphs with eigenvalues of multiplicity $1$ are isomorphic.
However, are there any known cospectral graphs with eigenvalues of multiplicity $1$ that are not isomorphic?
The only non-isomorphic graphs with same spectrum of known multiplicities I can think of are strongly regular graphs, and since these have exactly $3$ eigenvalues, they do not provide an answer to the above.