  We study in detail the possibility that the flat directions of the Minimal
Supersymmetric Standard Model (MSSM) could act as a curvaton and generate the
observed adiabatic density perturbations. For that the flat direction energy
density has to dominate the Universe at the time when it decays. We point out
that this is not possible if the inflaton decays into MSSM degrees of freedom.
If the inflaton is completely in the hidden sector, its decay products do not
couple to the flat direction, and the flat direction curvaton can dominate the
energy density. This requires the absence of a Hubble-induced mass for the
curvaton, e.g. by virtue of the Heisenberg symmetry. In the case of hidden
radiation, $n=9$ is the only admissible direction; for other hidden equations
of state, directions with lower $n$ may also dominate. We show that the MSSM
curvaton is further constrained severely by the damping of the fluctuations,
and as an example, demonstrate that in no-scale supergravity it would fragment
into $Q$ balls rather than decay. Damping of fluctuations can be avoided by an
initial condition, which for the $n=9$ direction would require an initial
curvaton amplitude of $\sim 10^{-2}M_p$, thereby providing a working example of
the MSSM flat direction curvaton.
