Suppose $\lambda$ is an non-orientable lamination on a closed orientable surface. How to construct an oriented double cover of $\lambda$?
the construction is the same as that for a manifold. to each point you associate the two orientations with the natural induced topology.if the connected lamination is orientable the construction has two components, otherwise only one.
this distinction, orientable or not, is interesting for laminations. for example, a one dimensional orientable lamination with cantor set transversals is the boundary of a two dimensional lamination with cantor set transversals.
I don't know whether or not a non orientable one dimensional lamination with cantor set transversals is the boundary of a non orientable two dimensional lamination.