  We study the stochastic heat equation with trace class noise and zero
Dirichlet boundary condition on a bounded polygonal domain O in R^2. It is
shown that the solution u can be decomposed into a regular part u_R and a
singular part u_S which incorporates the corner singularity functions for the
Poisson problem. Due to the temporal irregularity of the noise, both u_R and
u_S have negative L_2-Sobolev regularity of order s<-1/2 in time. The regular
part u_R admits spatial Sobolev regularity of order r=2, while the spatial
Sobolev regularity of u_S is restricted by r<1+\pi/\gamma, where \gamma is the
largest interior angle at the boundary of O. We obtain estimates for the
Sobolev norm of u_R and the Sobolev norms of the coefficients of the
singularity functions. The proof is based on a Laplace transform argument
w.r.t. the time variable. The result is of interest in the context of numerical
methods for stochastic PDEs.
