  A lot of attention has been drawn over the last few years by the
investigation of the geometry of spherical random eigenfunctions (random
spherical harmonics) in the high frequency regime, i.e ., for diverging
eigenvalues. In this paper, we present a review of these results and we collect
for the first time a comprehensive numerical investigation, focussing on
particular on the behaviour of Lipschitz-Killing curvatures/Minkowski
functionals (i.e., the area, the boundary length and the Euler-Poincar\'e
characteristic of excursion sets) and on critical points. We show in particular
that very accurate analytic predictions exist for their expected values and
variances, for the correlation among these functionals, and for the
cancellation that occurs for some specific thresholds (the variances becoming
an order of magnitude smaller - the so-called Berry's cancellation phenomenon).
Most of these functionals can be used for important statistical applications,
for instance in connection to the analysis of Cosmic Microwave Background data.
