If new physics (e.g. SUSY) does not show up as direct evidence at the
LHC, it could still be observable in FCNC processes involving the
$t$-quark. We take a close look at the process $t\to c + h/Z$ and show that
its branching ratio in the Standard Model is subject to three
mechanisms of suppression. To obtain an observable signal, one needs
to evade all these mechanisms in a theory beyond the Standard
Model. We show that a theory like the cMSSM cannot provide a big
enough enhancement. However, in a framework like $R$-parity-violating
SUSY, observable signals are a distinct possibility.