  The author has previously associated to each commutative ring with unit $R$
and \'etale groupoid $\mathscr G$ with locally compact, Hausdorff and totally
disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we
characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As
corollaries, we extend the characterization of Abrams, Aranda~Pino and
Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over
a field to arbitrary commutative coefficient rings and we recover the
characterization of Okni\'nski of Noetherian inverse semigroup algebras and of
Zelmanov of Artinian inverse semigroup algebras.
