  A digraph $H$ is called ubiquitous if every digraph that contains arbitrarily
many vertex-disjoint copies of $H$ also contains infinitely many
vertex-disjoint copies of $H$. We study oriented double rays, that is, digraphs
$H$ whose underlying undirected graphs are double rays. Calling a vertex of an
oriented double ray a turn if it has in-degree or out-degree 2, we prove that
an oriented double ray with at least one turn is ubiquitous if and only if it
has a (finite) odd number of turns. It remains an open problem to determine
whether the consistently oriented double ray is ubiquitous.
