In this talk, we study the stochastic heat equation on R^d driven by a multiplicative Gaussian noise which is white in time and colored in space. The diffusion coefficient rho can be degenerate, which includes the parabolic Anderson model rho(u)= u as a special case. The initial data is rough in the sense that it can be any measure, including the Dirac delta measure, that satisfies some mild integrability conditions. Under these degenerate conditions, for any given t>0 and distinct m points x_1, ... x_m in R^d, we establish the existence, regularity, and strict positivity of the joint density of the random vector (u(t,x_1), ...u(t,x_m)). The talk is based on a recent jointwork with Yaozhong Hu and David Nualart for the spatial dimension case, and an ongoing research project with Jingyu Huang for the higher spatial dimension case.