  We study a large class of stochastic $p$-Laplace Allen-Cahn equations with
singular potential. Under suitable assumptions on the (multiplicative-type)
noise we first prove existence, uniqueness, and regularity of variational
solutions. Then, we show that a random separation property holds, i.e. almost
every trajectory is strictly separated in space and time from the potential
barriers. The threshold of separation is random, and we further provide
exponential estimates on the probability of separation from the barriers.
Eventually, we exhibit a convergence-in-probability result for the random
separation threshold towards the deterministic one, as the noise vanishes, and
we obtain an estimate of the convergence rate.
