  We study a singular or degenerate equation in non-divergence form modeled by
the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N
u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$
regularity of viscosity solutions in the full range $\gamma>-1$ and $p>1$, and
provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to
2 and $\gamma$ is close to 0.
