Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2×2 canonical systems). We’ll present several Szego-type theorems for generalized indefinite strings and related spectral problems (including Krein strings and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert-Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa-Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform. The talk is based on joint work with J.Eckhardt.