  For a graph G, we define a small automorphism as one that maps some vertex
into its neighbour. We investigate the edge colourings of G that break every
small automorphism of G. We show that such a colouring can be chosen from any
set of lists of length three. In addition, we show that any set of lists of
length two on both edges and vertices of G yields a total colouring which
breaks all the small automorphisms of $G$. These results are sharp and they
match the non-list variants.
