The problem of turbulence spectra generated by the singularities located on lines and planes is considered. It is shown that the frequency spectrum of fluid-surface displacements due to whitecaps (linear singularities) is scaled like a weakly turbulent Zakharov-Filonenko spectrum. The corresponding wave-vector spectrum may be highly anisotropic with a decrease in maximum, as in the Phillips spectrum. However, in the isotropic situation, the spectrum differs markedly from the Phillips form. For a highly anisotropic two-dimensional turbulence, the vorticity jumps can generate the Kraichnan power-law distribution in the region of maximal angular peak. For the isotropic distribution, the turbulence spectrum coincides with the Saffman spectrum. For the shock-generated acoustic turbulence, the spectrum has the form of the Kadomtsev-Petviashvili spectrum E(ω) ∼ ω(-2) for all spatial dimensionalities.