  We consider ${\cal N}=4$ conformal supergravity with an arbitrary holomorphic
function of the complex scalar $S$ which parametrizes the $SU(1,1)/U(1)$ coset.
Assuming non-vanishings vevs for $S$ and the scalars in a symmetric matrix
$E_{ij}$ of the $\overline{\bf 10}$ of $SU(4)$ R-symmetry group, we determine
the vacuum structure of the theory. We find that the possible vacua are
classified by the number of zero eigenvalues of the scalar matrix and the
spacetime is either Minkowski, de Sitter or anti-de Sitter. We determine the
spectrum of the scalar fluctuations and we find that it contains tachyonic
states which however can be removed by appropriate choice of the unspecified at
the supergravity level holomorphic function. Finally, we also establish that
$S$-supersymmetry is always broken whereas $Q$-supersymmetry exists only on
flat Minkowski spacetime.
