  We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of
nonlinear stochastic heat equations with multiplicative noise:
\begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial
x^2} u(t,x)+\sigma(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$,
$x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time
white noise. We say that a point $z\in\mathbb{R}^d$ is polar if
\begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0.
\end{equation*} We show that in the critical dimension $d=6$, almost all points
in $\mathbb{R}^d$ are polar.
