  This paper studies the Sobolev regularity estimates of weak solutions of a
class of singular quasi-linear elliptic problems of the form $u_t -
\mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with
homogeneous Dirichlet boundary conditions over bounded spatial domains. Our
main focus is on the case that the vector coefficients $\mathbb{A}$ are
discontinuous and singular in $(x,t)$-variables, and dependent on the solution
$u$. Global and interior weighted $W^{1,p}(\Omega, \omega)$-regularity
estimates are established for weak solutions of these equations, where $\omega$
is a weight function in some Muckenhoupt class of weights. The results obtained
are even new for linear equations, and for $\omega =1$, because of the
singularity of the coefficients in $(x,t)$-variables
