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.