In this talk we will present two different eigenvalue problems arising in the theory of linear elasticity. In the first one we discuss Steklov-type eigenvalues, that is the spectral parameter only takes part in a Robin boundary condition. The second eigenvalue problem is the so called "Jones eigenproblem", in which we seek eigenpairs of the Lam\'e operator with traction boundary condition that are purely tangential to the boundary. We will discuss existence of these eigenpairs as well as numerical approximations of them.