A time-domain topological sensitivity (TS) approach is developed for elastic-wave imaging of media of arbitrary geometry. The TS, which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal hole, is expressed in terms of the time convolution of the free field and a supplementary adjoint field as a function of that specified location. Following previous studies performed under (mostly) static or time-harmonic conditions, the TS field is here considered as a natural and computationally efficient approach for defining a defect indicator function. This study emphasizes the implementation and exploitation of TS fields using standard displacement-based FEM approaches, a straightforward task once the correct sensitivity formulation is available. A comprehensive set of numerical experiments on 3-D and 2-D elastodynamic and acoustic configurations is reported, allowing to assess and highlight many features of the proposed TS-based fast qualitative identification such as its ability to identify multiple defects and its robustness against data noise.