  We are concerned with interior and global gradient estimates for solutions to
a class of singular quasilinear elliptic equations with measure data, whose
prototype is given by the $p$-Laplace equation $-\Delta_p u=\mu$ with $p\in
(1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in
\big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22],
respectively. In this paper, we improve the results in [22] and address the
open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global
modulus of continuity estimates of the gradients of solutions are also
established.
