We obtain sharp local C1,α\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\alpha }$$\end{document} regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by Δpu=γ(u-φ)γ-1in{u>φ},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Delta _p u=\gamma (u-\varphi )^{\gamma -1}\,\text { in }\,\{u>\varphi \}, \end{aligned}$$\end{document}for 0<γ<1\documentclass[12pt]{minimal}
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\begin{document}$$0<\gamma <1$$\end{document} and p≥2\documentclass[12pt]{minimal}
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\begin{document}$$p\ge 2$$\end{document}. At the free boundary ∂{u>φ}\documentclass[12pt]{minimal}
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\begin{document}$$\partial \{u>\varphi \}$$\end{document}, we prove optimal C1,τ\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\tau }$$\end{document} regularity of solutions, with τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document} given explicitly in terms of p, γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} and smoothness of φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}, which is new even in the linear setting.
