  In this paper we introduce Calder\'on-Zygmund theory for singular stochastic
integrals with operator-valued kernel. In particular, we prove
$L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse
domination and sharp weighted bounds are obtained under a Dini condition on the
kernel, leading to a stochastic version of the solution to the
$A_2$-conjecture. The results are applied to obtain $p$-independence and
weighted bounds for stochastic maximal $L^p$-regularity both in the complex and
real interpolation scale. As a consequence we obtain several new regularity
results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and
angular domains.
