  Each of several possible definitions of local injectivity for a homomorphism
of an oriented graph $G$ to an oriented graph $H$ leads to an injective
oriented colouring problem. For each case in which such a problem is solvable
in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such
that an oriented graph $G$ has an injective oriented colouring with the given
number of colours if and only if there is no $F \in \mathcal{F}$ for which
there is a locally-injective homomorphism of $F$ to $G$.
