  Given an infinite iterated function system (IFS) $\mathcal{F}$, we define its
dimension spectrum $D(\mathcal{F})$ to be the set of real numbers which can be
realised as the dimension of some subsystem of $\mathcal{F}$. In the case where
$\mathcal{F}$ is a conformal IFS, the properties of the dimension spectrum have
been studied by several authors. In this paper we investigate for the first
time the properties of the dimension spectrum when $\mathcal{F}$ is a
non-conformal IFS. In particular, unlike dimension spectra of conformal IFS
which are always compact and perfect (by a result of Chousionis, Leykekhman and
Urba\'{n}ski, Selecta 2019), we construct examples to show that
$D(\mathcal{F})$ need not be compact and may contain isolated points.
