  We perform a general study about the existence of non-supersymmetric minima
with vanishing cosmological constant in supergravity models involving only
chiral superfields. We study the conditions under which the matrix of second
derivatives of the scalar potential is positive definite. We show that there
exist very simple and strong necessary conditions for stability that constrain
the Kahler curvature and the ratios of the supersymmetry-breaking auxiliary
fields defining the Goldstino direction. We then derive more explicitly the
implications of these constraints in the case where the Kahler potential for
the supersymmetry-breaking fields is separable into a sum of terms for each of
the fields. We also discuss the implications of our general results on the
dynamics of moduli fields arising in string compactifications and on the
relative sizes of their auxiliary fields, which are relevant for the soft terms
of matter fields. We finally comment on how the idea of uplifting a
supersymmetric AdS vacuum fits into our general study.
