We study vector-valued solutions u(t, x) is an element of R-d to systems of nonlinear stochastic heat equations with multiplicative noise,

partial derivative/partial derivative t u(t, x) = partial derivative(2)/partial derivative x(2) u(t, x) + sigma (u(t, x)(W) over dot (t, x).

Here, t >= 0, x is an element of R and (W) over dot (t, x) is an R-d-valued space-time white noise. We say that a point z is an element of R-d is polar if

P{u(t, x) = z for some t > 0 and x is an element of R} = 0.

We show that, in the critical dimension d = 6, almost all points in R-d are polar.