New inversion algorithms have been developed for estimating the formation shear and borehole-fluid slownesses, using either the borehole Stoneley or dipole flexural dispersion in well-bonded cased boreholes surrounded by a fast or slow isotropic and purely elastic formation. Two inversion algorithms have been developed for each type of formation. The first algorithm inverts either the measured borehole Stoneley or flexural dispersion at select frequencies for the formation shear slowness when all other model parameters are known. The second algorithm inverts either the borehole Stoneley or flexural dispersion for both the formation shear and borehole-fluid compressional slownesses. Optimal bandwidths for the inversion of the Stoneley and dipole flexural dispersions for the formation shear slowness range from about 5 to 8 kHz. The well-bonded cased borehole dispersion sensitivity to formation shear slowness becomes larger at these higher frequencies than in an open-hole. Moreover, the Stoneley dispersion sensitivity to the borehole-fluid compressional slowness is so large that it becomes necessary to input an extremely accurate estimate of fluid compressional slowness in the inversion algorithm. Inverted formation shear slowness from the Stoneley data in a fast formation exhibits an uncertainty of about 3%, whereas the input borehole-fluid slowness has an uncertainty of 0.5%. Given a certain amount of uncertainty in the borehole-fluid slowness, one can then estimate possible variances in the inverted formation shear slowness. In contrast, inversion of the flexural dispersion for formation shear slowness is less sensitive to the input borehole-fluid compressional slowness in the preferred frequency band of 5 to 8 kHz. Inverted formation shear slownesses in slow formations that use either the Stoneley or flexural dispersion are also far less sensitive to uncertainties in the borehole-fluid compressional slowness.