  We consider very constrained versions of the minimal supersymmetric extension
of the Standard Model (VCMSSMs) which, in addition to constraining the scalar
masses m_0 and gaugino masses m_{1/2} to be universal at some input scale,
impose relations between the trilinear and bilinear soft supersymmetry breaking
parameters A_0 and B_0. These relations may be linear, as in simple minimal
supergravity models, or nonlinear, as in the Giudice-Masiero mechanism for
generating the Higgs-mixing mu term. We discuss the application of the
electroweak vacuum conditions in VCMSSMs, which may be used to make a
prediction for tan beta as a function of m_0 and m_{1/2} that is usually
unique. We baseline the discussion of the parameter spaces allowed in VCMSSMs
by updating the parameter space allowed in the CMSSM for fixed values of tan
beta with no relation between A_0 and B_0 assumed {\it a priori}, displaying
contours of B_0 for a fixed input value of A_0, incorporating the latest CDF/D0
measurement of m_t and the latest BNL measurement of g_mu - 2. We emphasize
that phenomenological studies of the CMSSM are frequently not applicable to
specific VCMSSMs, notably those based on minimal supergravity, which require
m_0 = m_{3/2} as well as A_0 = B_0 + m_0. We then display (m_{1/2}, m_0) planes
for selected VCMSSMs, treating in a unified way the parameter regions where
either a neutralino or the gravitino is the LSP. In particular, we examine in
detail the allowed parameter space for the Giudice-Masiero model.
