Wavefront set of characteristic function of rough set

It is a standard exercise to show that if $$X\subseteq\mathbb{R}^n$$ has smooth boundary, then the characteristic function $$1_X$$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}:\xi\text{ is normal to }\partial X\}.$$ This is shown by locally "flattening" to reduce to the case of the upper half space.

My question is what happens when $$X$$ does not have smooth boundary, such as the interior of a fractal curve. For example, if $$X$$ is the interior of the Koch snowflake, how would one compute the wavefront set of $$1_X$$?

Let me first begin with an elementary example, taking $$X=[0,1]^2$$ in $$\mathbb R^2$$. It is then easy to see directly that the wave-front-set of $$\mathbb 1_X$$ is everywhere the conormal bundle except at the four corners $$c_j$$: $$WF\mathbb 1_X=\cup_{1\le j\le 4}(c_j;\mathbb R^2\backslash {(0,0)})\cup\text{conormal bundle at the smooth points}.$$ To prove this, it is enough to check the wave-front-set of $$H(x_1)H(x_2)$$ with $$H=\mathbb 1_{[0,+\infty)}$$ which amounts to check the wave-front-set of $$\delta_0(x_1)\otimes\delta_0(x_2)$$ which contains at $$(0,0)$$ every direction since the Fourier transform is 1.
There are of course (much) more refined results. If you take a look at Lars Hörmander's ALPDO first volume, Springer Grundlehren 256 on page 300, you will find the definition of the normal set to any closed subset of $$\mathbb R^n$$ and Theorem 8.5.6' asserts that the analytic wave-front-set of a distribution $$u$$ contains the normal set to the support.
For the example you give, I believe that the $$C^\infty$$ wave-front-set is simply the product of the boundary with $$\mathbb R^2\backslash {(0,0)}$$. However for $$s$$ real, we may define the $$H^s$$ wave-front-set and I guess that the Hausdorff dimension of the fractal set you consider should be linked to the $$H^s$$ wave-front-set, in the sense that it should be trivial at some threshold $$s_0$$ and that $$s_0$$ should be linked to the Hausdorff dimension. The paper MR1277392, by Falcone, is addressing a related question.