  Strongly disordered superconducting films have been observed to undergo
finite temperature transitions to a superinsulating state, of apparently
infinite resistance, mirroring superconductivity. Approaching the transition,
some of the films reportedly exhibit Berezinskii-Kosterlitz-Thouless (BKT)
criticality implying that superinsulation is associated with an ordered charge
BKT phase. An even more singular Vogel-Fulcher-Tammann (VFT) criticality has
also been seen, positing the question of the existence of fundamentally
different states of finite temperature insulators. Here we develop a theory of
the criticality of a disordered lateral Josephson junction array with weak
Josephson coupling. We show that it is equivalent to a two-dimensional Coulomb
gas with logarithmically correlated disorder. We show that strong disorder
results in a regime exhibiting VFT criticality instead of the usual BKT one,
and find that it corresponds to transition to a nonergodic insulator phase.
