  The main objects of this paper include some degenerate and nonlocal elliptic
operators which naturally arise in the conformal invariant theory of
Poincar\'e-Einstein manifolds. These operators generally reflect the
correspondence between the Riemannian geometry of a complete
Poincar\'e-Einstein manifold and the conformal geometry of its associated
conformal infinity. In this setting, we develop the quantitative
differentiation theory that includes quantitative stratification for the
singular set and Minkowski type estimates for the (quantitatively) stratified
singular sets. All these, together with a new $\epsilon$-regularity result for
degenerate/singular elliptic operators on Poincar\'e-Einstein manifolds, lead
to uniform Hausdorff measure estimates for the singular sets. Furthermore, the
main results in this paper provide a delicate synergy between the geometry of
Poincar\'e-Einstein manifolds and the elliptic theory of associated degenerate
elliptic operators.
