  Subspace codes have recently been used for error correction in random network
coding. In this work, we focus on one-orbit cyclic subspace codes. If $S$ is an
$\mathbb{F}_q$-subspace of $\mathbb{F}_{q^n}$, then the one-orbit cyclic
subspace code defined by $S$ is \[ \mathrm{Orb}(S)=\{\alpha S \colon \alpha \in
\mathbb{F}_{q^n}^*\}, \]where $\alpha S=\lbrace \alpha s \colon s\in S\rbrace$
for any $\alpha\in \mathbb{F}_{q^n}^*$. Few classification results of subspace
codes are known, therefore it is quite natural to initiate a classification of
cyclic subspace codes, especially in the light of the recent classification of
the isometries for cyclic subspace codes. We consider three-dimensional
one-orbit cyclic subspace codes, which are divided into three families: the
first one containing only $\mathrm{Orb}(\mathbb{F}_{q^3})$; the second one
containing the optimum-distance codes; and the third one whose elements are
codes with minimum distance $2$. We study inequivalent codes in the latter two
families.
