We consider the stochastic heat equation with a multiplicative space-time white noise forcing term under standard "intermitency conditions.” The main byproduct of this talk is that, under mild regularity hypotheses, the a.s.-boundedness of the solution$x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:=\lim_{|x|\to\infty} \vert\log u_0(x)\vert/(\log|x|)^{2/3}$.