  We compare light propagation through an intense electromagnetic background as
described by three different nonlinear electrodynamics: Born-Infeld (BI),
Euler-Heisenberg (EH), and Modified Maxwell (MM). We use the concept of
effective metric to determine the phase velocities of a propagating wave from
the BI and EH nonlinear electrodynamics and use them to set constraints on the
MM nonlinear parameter. In a second part of the paper, we consider the black
hole solutions of the nonlinear electrodynamics coupled with General Relativity
and determine the shadows cast by the static black holes. Confronting the
observations of the shadows of M87$^*$ and Sagittarius A$^*$ with our
theoretical results allows us to set restrictions on the nonlinear parameters.
