  This paper analyzes a full discretization of a three-dimensional stochastic
Allen-Cahn equation with multiplicative noise. The discretization uses the
Euler scheme for temporal discretization and the finite element method for
spatial discretization. A key contribution of this work is the introduction of
a novel stability estimate for a discrete stochastic convolution, which plays a
crucial role in establishing pathwise uniform convergence estimates for fully
discrete approximations of nonlinear stochastic parabolic equations. By using
this stability estimate in conjunction with the discrete stochastic maximal
$L^p$-regularity estimate, the study derives a pathwise uniform convergence
rate that encompasses general general spatial $L^q$-norms. Moreover, the
theoretical convergence rate is verified by numerical experiments.
