  In this paper we present a mathematical and numerical analysis of an
eigenvalue problem associated to the elasticity-Stokes equations stated in two
and three dimensions. Both problems are related through the Herrmann pressure.
Employing the Babu\v ska--Brezzi theory, it is proved that the resulting
continuous and discrete variational formulations are well-posed. In particular,
the finite element method is based on general inf-sup stables pairs for the
Stokes system, such that, Taylor--Hood finite elements. By using a general
approximation theory for compact operators, we obtain optimal order error
estimates for the eigenfunctions and a double order for the eigenvalues. Under
mild assumptions, we have that these estimates hold with constants independent
of the Lam\'e coefficient $\lambda$. In addition, we carry out the reliability
and efficiency analysis of a residual-based a posteriori error estimator for
the spectral problem. We report a series of numerical tests in order to assess
the performance of the method and its behavior when the nearly incompressible
case of elasticity is considered.
