Stochastic partial differential equations with polynomial coefficients have many applications in the study of spatially distributed populations in genetics, epidemiology, ecology, and chemical kinetics. The purpose of this paper is to describe some methods for investigating the qualitative behavior of spatially homogenous solutions of nonlinear stochastic partial differential equations. The stability and bifurcation of solutions, as well as the behavior near critical points are discussed. The methods employed are the nonlinear Markov approximation, moment density inequalities for identifying invariant sets of behaviors and rescaling and quasi-linearization around critical points.