We give a characterization of $G$-regularity for super-Brownian
motion and the Brownian snake. More precisely, we define a capacity on $E =
(0,\infty) \times\mathbb{R}^d$, which is not invariant by translation. We then
prove that the measure of hitting a Borel set $A \subset E$ for the graph of
the Brownian snake excursion starting at (0,0) is comparable, up to
multiplicative constants, to its capacity. This implies that super-Brownian
motion started at time 0 at the Dirac mass $\varrho_0$ hits immediately $A$
[i.e., (0,0) is $G$ -regular for $A_c$] if and only if its capacity is
infinite. As a direct consequence, if $Q \subset E$ is a domain such that (0,0)
$\in\varrho Q$, we give a necessary and sufficient condition for the existence
on $Q$ of a positive solution of $\varrho_tu + 1/2\Delta u = 2u^2$, which blows
up at (0,0). We also give an estimate of the hitting probabilities for the
support of super-Brownian motion at fixed time. We prove that if $d\geq 2$, the
support of super-Brownian motion is intersection-equivalent to the range of
Brownian motion.