  We prove a generalized Kac-Rice formula that, in a well defined regular
setting, computes the expected cardinality of the preimage of a submanifold via
a random map, by expressing it as the integral of a density. Our proof starts
from scratch and although it follows the guidelines of the standard proofs of
Kac-Rice formula, it contains some new ideas coming from the point of view of
measure theory. Generalizing further, we extend this formula to any other type
of counting measure, such as the intersection degree.
  We discuss in depth the specialization to smooth Gaussian random sections of
a vector bundle. Here, the formula computes the expected number of points where
the section meets a given submanifold of the total space, it holds under
natural non-degeneracy conditions and can be simplified by using appropriate
connections. Moreover, we point out a class of submanifolds, that we call
sub-Gaussian, for which the formula is locally finite and depends continuously
with respect to the covariance of the first jet. In particular, this applies to
any notion of singularity of sections that can be defined as the set of points
where the jet prolongation meets a given semialgebraic submanifold of the jet
space.
  Various examples of applications and special cases are discussed. In
particular, we report a new proof of the Poincar\'e kinematic formula for
homogeneous spaces and we observe how the formula simplifies for isotropic
Gaussian fields on the sphere.
