We construct a simple topological invariant of certain $3$-manifolds, including quotients of $S^3$ by finite groups, based on the fact that the tangent bundle of an orientable $3$-manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We also use properties of this invariant to show that there is an oriented $3$-manifold with no universally tight contact structure. We generalise and sharpen this invariant to an invariant of a finite covering of a $3$-manifold.