  For the obstacle problem involving a convex fully nonlinear elliptic
operator, we show that the singular set in the free boundary stratifies. The
top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower
strata are covered by $C^{1,\log^\varepsilon}$-manifolds. This essentially
recovers the regularity result obtained by Figalli-Serra when the operator is
the Laplacian.
