  In this article, we give probabilistic versions of Sobolev embeddings on any
Riemannian manifold $(M,g)$. More precisely, we prove that for natural
probability measures on $L^2(M)$, almost every function belong to all spaces
$L^p(M)$, $p<+\infty$. We then give applications to the study of the growth of
the $L^p$ norms of spherical harmonics on spheres $\mathbb{S}^d$: we prove
(again for natural probability measures) that almost every Hilbert base of
$L^2(\mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly
bounded in all $L^p(\mathbb{S}^d), p<+\infty$ spaces. We also prove similar
results on tori $\mathbb{T}^d$. We give then an application to the study of the
decay rate of damped wave equations in a frame-work where the geometric control
property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated
for a measure 0 set of trajectories, we prove that there exists almost surely a
rate. Finally, we conclude with an application to the study of the
$H^1$-supercritical wave equation, for which we prove that for almost all
initial data, the weak solutions are strong and unique, locally in time.
