This paper provides a canonical compactification of the plane {\mathbb{R}}^{2} by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather (1982). Moreover, any homeomorphism of {\mathbb{R}}^{2} preserving the foliations extends on the circle at infinity. Then, this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on {\mathbb{R}}^{2} preserving one foliation or two transverse foliations. In particular, the action on the circle at infinity associated to an Anosov flow X on a closed 3 -manifold is minimal if and only if X is non- {\mathbb{R}} -covered.