In any approach to calculation of the wall pressure spectrum of a TBL based on an inhomogeneous (Lighthill) wave equation, singularities of the spectral density arise along various lines or surfaces in (k,ω) space. These lines of surfaces correspond to the acoustic or structural dispersion relations, and the singularities may be controlled by such mechanisms as surface finiteness, dissipation in the structure or fluid, and surface compliance. This paper will examine the way in which these various mechanisms control the singularities in different regimes of the spatial and dynamical parameters involved, and will give scaling laws for the pressure spectral densities at the critical conditions in (k,ω) space. [Work supported by ONR, Code 425.]