Generic properties of elastic phonon transport at a disordered interface are studied. The results show that phonon transmittance is a strong function of frequency and the disorder correlation length. At frequencies lower than the van Hove singularity the transmittance at a given frequency increases as the correlation length decreases. At low frequencies, this is reflected by different power laws for phonon conductance across correlated and uncorrelated disordered interfaces which are in approximate agreement with the perturbation theory of an elastic continuum. These results can be understood in terms of simple mosaic and two-color models of the interface.