  We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular
obstacle problems, Euler-Lagrange equation of which is given by
  $$
  \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\},
  $$
  for $0<\gamma<1$ and $p\ge2$. At the free boundary $\partial\{u>\varphi\}$,
we prove optimal $C^{1,\tau}$ regularity of solutions, with $\tau$ given
explicitly in terms of $p$, $\gamma$ and smoothness of $\varphi$, which is new
even in the linear setting.
