What's the regularity of the level set of a ''semi-nondegenerate" smooth function on closed manifold?

Question

Let $(M^2,g)$ be a closed smooth Riemannian manifold and $f$ be a smooth function on $M^2$. Suppose $A=f^{-1}(0)\neq\emptyset$. And assume that if $f(x) =0$ and $\nabla f(x)=0$, then $\Delta f(x)>0$. What is the regulariy of $A$? Obviously, $A$ is closed and has no interior point. Can we say something more? For example, what is largest Hausdorff dimension can $A$ have? Can $A$ be a union of several arcs and isolated points? A different but relavent equsion is The regularity of intersection of a minimal surface and a surface of positive mean curvature?.