  Horizontal points of smooth submanifolds in stratified groups play the role
of singular points with respect to the Carnot-Carathe'odory distance. When we
consider hypersurfaces, they coincide with the well known characteristic
points. In two step groups, we obtain pointwise estimates for the Riemannian
surface measure at all horizontal points of $C^{1,1}$ smooth submanifolds. As
an application, we establish an integral formula to compute the spherical
Hausdorff measure of any $C^{1,1}$ submanifold. Our technique also shows that
$C^2$ smooth submanifolds everywhere admit an intrinsic blow-up and that the
limit set is an intrinsically homogeneous algebraic variety.
