  In supersymmetric extensions of the Standard Model of particle physics (SM),
goldstino superpartners --- scalar and pseudoscalar sgoldstinos --- can be
light enough for emerging in decays of SM particles. Sgoldstino interaction
with SM fields is suppressed by the scale of supersymmetry breaking in the
whole theory. Hence, searches for sgoldstinos give an opportunity to probe the
underlying mechanism of supersymmetry breaking. Sgoldstino couplings to SM
fields are proportional to the supersymmetry breaking parameters --- MSSM soft
terms --- and therefore can lead to flavor violating processes in quark and
lepton sectors. We consider flavor violating processes involving sgoldstino
pair production which are driven by sgoldstino couplings proportional to squark
and slepton soft mass terms, \tilde{m^2_{LL}} and \tilde{m^2_{RR}}. We find
that present limits on off-diagonal entries in squark and slepton squared mass
matrices allow t-, b-, c-quark and \tau-lepton decays at levels available for
study with existing data (BaBar, Belle, CLEOc) and in ongoing experiments
(LHCb, CMS, ATLAS). In particular, we obtain the following branching ratios
Br(t--> c SP)\lesssim 10^{-7}, Br(\tau--> \mu SP) \lesssim 10^{-7}, Br(B_s-->
SP) \lesssim 10^{-4}, Br(B--> K^{(*)} SP) \lesssim 10^{-4}, Br(D--> SP)
\lesssim 10^{-7} with sgoldstino subsequent decays into kinematically allowed
pairs of SM particles \gamma\gamma, e^+e^-, \mu^+\mu^-, etc. Remarkably, the
prominent signature of sgoldstino pair production is two muon pairs with pair
momenta peaked at sgoldstino masses.
