Gravitational waves (GW) from compact binary coalescences are natural cosmological probes. On the one hand, these standard siren sources provide direct measurements of luminosity distance. In addition, features in the mass distribution, such as mass gaps or peaks, will redshift, and thus provide independent constraints on their redshift distribution. On the other hand, GWs from high-redshifts could be gravitationally lensed, mapping the matter distribution in the Universe. In this talk I will explore the use of the mass spectrum of neutron stars and black holes in GW compact binary sources to probe the standard cosmological model. I will show the potential of these “spectral sirens” to constrain the high-redshift expansion rate and test gravity, demonstrating that degeneracies between redshift evolution of the source masses and cosmology can be broken. I will then discuss the prospects and challenges of identifying lensed GWs.