  We consider sample path properties of the solution to the stochastic heat
equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a
L\'evy space-time white noise. When viewed as a stochastic process in time with
values in an infinite-dimensional space, the solution is shown to have a
c\`adl\`ag modification in fractional Sobolev spaces of index less than $-\frac
d 2$. Concerning the partial regularity of the solution in time or space when
the other variable is fixed, we determine critical values for the
Blumenthal-Getoor index of the L\'evy noise such that noises with a smaller
index entail continuous sample paths, while L\'evy noises with a larger index
entail sample paths that are unbounded on any non-empty open subset. Our
results apply to additive as well as multiplicative L\'evy noises, and to
light- as well as heavy-tailed jumps.
