= Solution
Let $u,v\in\mathcal D'(\mathbb R^n)$, with at least one a <compactly supported distribution>, so their <convolution> is defined. The <support and singular support rules for convolution> are
$$
\boxed{\operatorname{supp}(u*v)\subseteq\operatorname{supp}u+\operatorname{supp}v,\qquad
\operatorname{sing\,supp}(u*v)\subseteq
\operatorname{sing\,supp}u+\operatorname{sing\,supp}v.}
$$
Here $A+B=\{a+b:a\in A,\ b\in B\}$ is the <Minkowski sum>. The compact-factor hypothesis also ensures that these sums of closed sets are closed. More generally, the same rules hold when addition is a <proper map> on $\operatorname{supp}u\times\operatorname{supp}v$, which is another sufficient condition for defining the <convolution>. The second inclusion explains why a compactly supported <smooth function> smooths every <distribution> under <convolution>.
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