  We introduce a notion of local level spacings and study their statistics
within a random-matrix-theory approach. In the limit of infinite-dimensional
random matrices, we determine universal sequences of mean local spacings and of
their ratios which uniquely identify the global symmetries of a quantum system
and its internal -- chaotic or regular -- dynamics. These findings, which offer
a new framework to monitor single- and many-body quantum systems, are
corroborated by numerical experiments performed for zeros of the Riemann zeta
function, spectra of irrational rectangular billiards and many-body spectra of
the Sachdev-Ye-Kitaev Hamiltonians.
