I've been trying to show that the wave front set for the characteristic function of the open ball, $B(0,1)$, is given by the boundary normal vectors $\{(x,\xi) \in S^1 \times \mathbb{R}^2-\{0\}$: $x \|\xi\}$ (i.e. $t\xi = x\in S^1$, $t\in \mathbb{R}-\{0\}$). I'd be very happy if someone showed me how to prove the statement.
Definition of the wave front set: https://en.wikipedia.org/wiki/Wave_front_set
On
you'll find all the necessary informations related to these kinds of characteristic functions and some general technics to calculate wave front set in
https://www.icts.res.in/sites/default/files/Jan_Boman_Lecture_Notes_0.pdf
Besides, this article is very easy to read. The second part of the document is related to the interplay between Radon transform and wave front set, but though it is worth reading, it is far form your subject
This is shown in detail in Section 4 of this paper. The idea is to transform to coordinates in terms of which the characteristic function of the ball is a tensor product between a smooth function and a one-dimensional Heaviside distribution. Of course, in this case polar coordinates do the trick.