  We establish a hierarchy of weighted majorization relations for the
singularities of generalized Lam\'e equations and the zeros of their Van Vleck
and Heine-Stieltjes polynomials as well as for multiparameter spectral
polynomials of higher Lam\'e operators. These relations translate into natural
dilation and subordination properties in the Choquet order for certain
probability measures associated with the aforementioned polynomials. As a
consequence we obtain new inequalities for the moments and logarithmic
potentials of the corresponding root-counting measures and their weak-$^*$
limits in the semi-classical and various thermodynamic asymptotic regimes. We
also prove analogous results for systems of orthogonal polynomials such as
Jacobi polynomials.
