I would like to construct a distribution $u$ on $\mathbb{R}^n$, for $n \geq 2$, for which the wavefront set consists of a single point on the sphere, i.e. for which $$\mathrm{WF}(u) = \{(0, t\xi) \ \mid \ t > 0 \}, $$ for some $\xi \in \mathbb{R}^n \setminus \{0\}$.
The wavefront set of a distribution $u \in \mathscr{D}'(\mathbb{R}^n)$ is defined as follows: we say that $(x_0, \xi_0) \in \mathbb{R}^n \times (\mathbb{R}^n \setminus \{0\})$ is not in the wavefront set of $u$ if and only if there exists some $\phi \in C_c^\infty(\mathbb{R}^n)$ with $\phi(x_0) \neq 0$ and some $\varepsilon > 0$ such that for every $\xi \in \mathbb{R}^n$ with $\|\xi\| \geq 1$ and $$\left\| \frac{\xi}{\|\xi\|} - \frac{\xi_0}{\|\xi_0\|} \right\| < \varepsilon, $$ we have that, for every positive integer $N$, $$|\mathcal{F}(\phi u)(\xi)| \leq C_N \|\xi\|^{-N}, $$ for some $C_N > 0$, where $\mathcal{F}$ is the Fourier transform. Note that, by Paley-Wiener theorem, $\mathcal{F}(\phi u)$ is indeed a function in the classical sense.
I am asking specifically for the case $n \geq 2$, as for the case $n = 1$ we can consider the distributions $(x \pm i0)^{-1}$, and their wavefront sets are computed here: wavefront sets of distributions.
I am not sure how to generalize the distribution $(x + i0)^{-1}$ on $\mathbb{R}^n$. We certainly need a distribution that is not real-valued, but I do not know how to choose it.