To an element in the completion of the set of Lagrangians for the spectral distance we associate a support.  We show that such a support is γ-coisotropic (a notion we shall define in the talk) and we shall give examples and counterexamples of γ-coisotorpic sets that can be (or cannot be) γ-supports.  Finally, we give some applications of these notions to singular support of sheaves (joint work with S. Guillermou) and dissipative dynamics, allowing us to extend the notion of Birkhoff attractor (joint with V. Humilière).