This work deals with theoretical and numerical aspects related to the behavior of the Steklov-Lamé eigenvalues on variable domains. After establishing the eigenstructure for the disk, we prove that for a certain class of Lamé parameters, the disk maximizes the first non-zero eigenvalue under area or perimeter constraints in dimension two. Upper bounds for these eigenvalues can be found in terms of the scalar Steklov eigenvalues, involving various geometric quantities. We prove that the Steklov-Lamé eigenvalues are upper semicontinuous for the complementary Hausdorff convergence of ε-cone domains and, as a consequence, there exist shapes maximizing these eigenvalues under convexity and volume constraints. A numerical method based on fundamental solutions is proposed for computing the Steklov-Lamé eigenvalues, allowing to study numerically the shapes maximizing the first ten non-zero eigenvalues.