  Introducing a general class of one-dimensional single-file systems (meaning
that particle crossings are prohibited) of interacting hardcore particles with
internal degrees of freedom (called charge), we exhibit a novel type of
dynamical universality reflected in anomalous statistical properties of
macroscopic fluctuating observables such as charge transfer. We find that
stringent dynamical constraints lead to universal anomalous statistics of
cumulative charge currents manifested both on the timescale characteristic of
typical fluctuations and also in the rate function describing rare events. By
computing the full counting statistics of net transferred charge between two
extended subsystems, we establish a number of unorthodox dynamical properties
in an analytic fashion. Most prominently, typical fluctuations in equilibrium
are governed by a universal distribution that markedly deviates from the
expected Gaussian statistics, whereas large fluctuations are described by an
exotic large-deviation rate function featuring an exceptional triple critical
point. Far from equilibrium, competition between dynamical phases leads to
dynamical phase transitions of first and second order and spontaneous breaking
of fluctuation symmetry of the univariate charge large-deviation function. The
rich phenomenology of the outlined dynamical universality is exemplified on an
exactly solvable classical cellular automaton of charged hardcore particles. We
determine the dynamical phase diagram in the framework of Lee-Yang's theory of
phase transitions and exhibit a hyper-dimensional diagram of distinct dynamical
regimes. Our findings lead us to conclude that the conventional classification
of dynamical universality classes based on the algebraic dynamical exponents
and asymptotic scaling functions that characterize hydrodynamic relaxation of
the dynamical structure factor is incomplete and calls for refinement.
