In this article we introduce a stochastic counterpart of the Hörmander condition and CalderĂłn-Zygmund theorem. Let W t W_t be a Wiener process in a probability space Ω \Omega and let K ( Ï , r , t , x , y ) K(\omega ,r,t,x,y) be a random kernel which is allowed to be stochastically singular in a domain O â R d \mathcal {O} \subset \mathbf {R}^d in the sense that E | â« 0 t â« | x â y | > Δ | K ( Ï , s , t , y , x ) | d y d W s | p = â â t , p , Δ > 0 , x â O . \begin{equation*} \mathbb {E} \left |\int _0^{t} \int _{|x-y|>\varepsilon }|K(\omega , s, t,y,x)|dy dW_s\right |^p = \infty \quad \forall \, t, p,\varepsilon >0,\, x\in \mathcal {O}. \end{equation*} We prove that the stochastic integral operator of the type T g ( t , x ) â â« 0 t â« O K ( Ï , s , t , y , x ) g ( s , y ) d y d W s \begin{align} \mathbb {T} g(t,x) \coloneq \int _0^{t} \int _{\mathcal {O}} K(\omega ,s,t,y,x) g(s,y)dy dW_s \end{align} is bounded on L p = L p ( Ω Ă ( 0 , â ) ; L p ( O ) ) \mathbb {L}_p=L_p \left (\Omega \times (0,\infty ); L_{p}(\mathcal {O}) \right ) for all p â [ 2 , â ) p \in [2,\infty ) if it is bounded on L 2 \mathbb {L}_2 and the following (which we call stochastic Hörmander condition) holds: there exists a quasi-metric Ï \rho on ( 0 , â ) Ă O (0,\infty )\times \mathcal {O} and a positive constant C 0 C_0 such that for X = ( t , x ) , Y = ( s , y ) , Z = ( r , z ) â ( 0 , â ) Ă O X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty ) \times \mathcal {O} , sup Ï â Ω , X , Y â« 0 â [ â« Ï ( X , Z ) â„ C 0 Ï ( X , Y ) | K ( r , t , z , x ) â K ( r , s , z , y ) | d z ] 2 d r > â . \begin{equation*} \sup _{\omega \in \Omega ,X,Y}\int _{0}^\infty \left [ \int _{\rho (X,Z) \geq C_0 \rho (X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right ]^2 dr >\infty . \end{equation*} Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp L p L_p -regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.