  We prove refined space-time regularity for the classical stochastic
Allen-Cahn equation with logarithmic potential. This allows to establish a
random separation property, i.e. that the trajectories of the solution are
strictly separated from the potential barriers. The present contribution
extends the results obtained in Bertacco et al. (2022), where separation was
proved only for $p$-Laplace operators with $p$ greater than the
space-dimension.
