  We study the FCNC top quark decays t -> c h in the framework of the MSSM,
where h= h^0,H^0,A^0 is any of the supersymmetric neutral Higgs bosons. We
include the leading set of SUSY-QCD and SUSY electroweak contributions. While
the FCNC top quark decay into the SM Higgs boson has such a negligible rate
that will not be accessible to any presently conceivable accelerator, we find
that there is a chance that the potential rates in the MSSM can be measured at
the high luminosity colliders round the corner, especially at the LHC and
possibly at a future LC, but we deem it difficult at the upgraded Tevatron. In
view of the large SUSY-QCD effects that we find in the Higgs channels, and due
to some discrepancies in the literature, we have revisited the FCNC top quark
decay into gluon, t -> c g, in our framework. We confirm that the possibility
of sizeable rates does not necessarily require a general pattern of
gluino-mediated FCNC interactions affecting both the LH and the RH sfermion
sectors -- the LH one being sufficient. However, given the present bounds on
sparticle masses, the gluon channel turns out to lie just below the expected
experimental sensibility, so our general conclusion is that the Higgs channels
t -> c h (especially the one for the light CP-even Higgs) have the largest
potential top quark FCNC rates in the MSSM, namely of order 10^-4.
