We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be chosen from any set of lists associated to the edges of a graph G, whenever the size of each list is at least $Δ-1$, where $Δ$ is the maximum degree of G, apart from a few exceptions. This holds both for finite and infinite graphs. The bound is optimal for every $Δ\ge 3$, and it is the same as in the non-list version.