The metric signature topological transitions associated with the propagation of electromagnetic waves in a dispersive metamaterial with frequency-dependent and anisotropic dielectric and magnetic responses are examined in the present work. The components of the reciprocal-space metric tensor depend upon both the electric permittivity and magnetic permeability of the metamaterial, which are taken as Drude-like dispersive models. A thorough study of the frequency dependence of the metric tensor is presented which leads to the possibility of topological transitions of the isofrequency surface determining the wave dynamics inside the medium, to a diverging photonic density of states at some range of frequencies, and to the existence of large wave vectors' modes propagating through the metamaterial.