  Amenability of any of the algebras described in the title is known to force
them to be finite-dimensional. The analogous problems for \emph{approximate}
amenability have been open for some years now. In this article we give a
complete solution for the first two classes, using a new criterion for showing
that certain Banach algebras without bounded approximate identities cannot be
approximately amenable. The method also provides a unified approach to existing
non-approximate amenability results, and is applied to the study of certain
commutative Segal algebras.
  Using different techniques, we prove that \emph{bounded} approximate
amenability of the second dual of a Fourier algebra implies that it is
finite-dimensional. Some other results for related algebras are obtained.
