  Weighted good-$\lambda$ type inequalities and Muckenhoupt-Wheeden type bounds
are obtained for gradients of solutions to a class of quasilinear elliptic
equations with measure data. Such results are obtained globally over
sufficiently flat domains in $\mathbb{R}^n$ in the sense of Reifenberg. The
principal operator here is modeled after the $p$-Laplacian, where for the first
time singular case $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$ is considered. Those
bounds lead to useful compactness criteria for solution sets of quasilinear
elliptic equations with measure data. As an application, sharp existence
results and sharp bounds on the size of removable singular sets are deduced for
a quasilinear Riccati type equation having a gradient source term with linear
or super-linear power growth.
