  We consider quantum scattering of particles in media exhibiting strong
dispersion degeneracy. In particular, we study flat-banded lattices and
linearly dispersed energy bands. The former constitute a prime example of
single-particle frustration while the latter show degeneracy at the few- and
many-particle level. We investigate both impurity and two-body scattering and
show that, quite generally, scattering does not occur, which we relate to the
fact that transition matrices vanish on the energy shell. We prove that
scattering is instead replaced by projections onto band-projected eigenstates
of the interaction potential. We then use the general results to obtain
localised flat band states that are eigenstates of impurity potentials with
vanishing eigenvalues in one-dimensional flat bands and study the particular
case of a sawtooth lattice. We also uncover the relation between certain
solutions of one-dimensional systems that have been categorised as "strange",
and the scattering states in linearly dispersed continuum systems.
