  We characterize the solvable Lie groups of the form ${\mathbb R}^m\rtimes
{\mathbb R}$, whose $C^*$-algebras are quasidiagonal. Using this result, we
determine the connected simply connected solvable Lie groups of type~I whose
$C^*$-algebras are strongly quasidiagonal. As a by-product, we give also
examples of amenable Lie groups with non-quasidiagonal $C^*$-algebras.
