  We perform measurements of phase-slip-induced switching current events on
different types of superconducting weak links and systematically study
statistical properties of the switching current distributions. We employ two
types of devices in which a weak link is formed either by a superconducting
nanowire or by a graphene flake subject to proximity effect. We demonstrate
that, independently on the nature of the weak link, higher moments of the
distribution take universal values. In particular, the third moment (skewness)
of the distribution is close to -1 both in thermal and quantum regimes. The
fourth moment (kurtosis) also takes a universal value close to 5. The
discovered universality of skewness and kurtosis is confirmed by an analytical
model. Our numerical analysis shows that introduction of extraneous noise into
the system leads to significant deviations from the universal values. We
suggest to use the discovered universality of higher moments as a robust tool
for checking against undesirable effects on noise in various types of
measurements.
