  We develop a stochastic analysis for a Gaussian process $X$ with singular
covariance by an intrinsic procedure focusing on several examples such as
covariance measure structure processes, bifractional Brownian motion, processes
with stationary increments. We introduce some new spaces associated with the
self-reproducing kernel space and we define the Paley-Wiener integral of first
and second order even when $X$ is only a square integrable process continuous
in $L^2$. If $X$ has stationary increments, we provide necessary and sufficient
conditions so that its paths belong to the self-reproducing kernel space. We
develop Skorohod calculus and its relation with symmetric-Stratonovich type
integrals and two types of It\^o's formula. One of Skorohod type, which works
under very general (even very singular) conditions for the covariance; the
second one of symmetric-Stratonovich type, which works, when the covariance is
at least as regular as the one of a fractional Brownian motion of Hurst index
equal to $H = \frac{1}{4}$.
