I would like to gain intuition how "singular" a distribution can be.
Let $X \in \mathbb R^n$ be open, $\mathcal D(X)$ be the smooth test functions on $X$ with canonical LF topology, and let $\mathcal D'(X)$ be the topological dual of $D(X)$.
Let $\Psi \in D'(X)$. Can the singular support of $ \Psi$, i.e. the complement in $X$ of the largest open subset of $X$, on which $\Psi$ is a locally integrable function, be open?
For example, the Dirac delta has singular support in $0$. Similarly, integrals with respect to Hausdorff measures on finite measure submanifolds of $X$ may be distributions, whose singular support is not open.
All examples of distributions I know constitute of a function "disturbed" by a "small" singularity.