  We introduce and analyze the characteristic foliation induced by a contact
structure on a branched surface, in particular a branched standard spine of a
3-manifold. We extend to (fairly general) singular foliations of branched
surfaces the local existence and uniqueness results which hold for genuine
surfaces. Moreover we show that global uniqueness holds when restricting to
tight structures. We establish branched versions of the elimination lemma. We
prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing
that branched spines can be used to construct contact structures in a given
homotopy class of plane fields.
