Let $u\in W^{1,2}(\Omega)$ where $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary. Moreover, we have $0\leq u\leq 1$.
Let set $\Gamma$ be defined as $$ \Gamma:=\{x\in\Omega,\,\, u(x)=0\} $$ Then, I am wondering that, what kind of regularity properties of $\Gamma$ can I have? May I have a subset $\Gamma'\subset \Gamma$ such that $\Gamma'$ is $\mathcal H^{1}$ rectifable? or even a $C^1$ curve?
For every closed set $\Gamma\subset \Omega$ there is a function $u$ that satisfies the stated requirements, namely $$ u(x) = \min(\operatorname{dist}(x,\Gamma), 1) $$ Consequently, there is nothing you can say about $\Gamma$ other than it's closed with respect to $\Omega$.