  We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin
spaces on rough sets. Our main results, stated in the simplest Sobolev space
setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$,
$\mathcal{D}(\Omega)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\,
u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue
measure and $\Omega$ is "thick" (in the sense of Triebel); and (ii) for a
$d$-set $\Gamma\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm
supp}\, u\subset \Gamma\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm
supp}\, u\subset \Gamma\}$ whenever $-\frac{n-d}{2}-m-1<s_{2}\leq
s_{1}<-\frac{n-d}{2}-m$ for some $m\in\mathbb N_0$. For (ii), we provide
concrete examples, for any $m\in\mathbb N_0$, where density fails when $s_1$
and $s_2$ are on opposite sides of $-\frac{n-d}{2}-m$. The results (i) and (ii)
are related in a number of ways, including via their connection to the question
of whether $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}=\{0\}$ for a
given closed set $\Gamma\subset\mathbb R^n$ and $s\in \mathbb R$. They also
both arise naturally in the study of boundary integral equation formulations of
acoustic wave scattering by fractal screens. We additionally provide analogous
results in the more general setting of Besov and Triebel--Lizorkin spaces.
