  For a Dawson-Watanabe superprocess $X$ on $\mathbb{R}^d$, it is shown in
Perkins (1990) that if the underlying spatial motion belongs to a certain class
of L\'evy processes that admit jumps, then with probability one the closed
support of $X_t$ is the whole space for almost all $t>0$ before extinction, the
so-called ``instantaneous propagation'' property. In this paper for
superprocesses on $\mathbb{R}^1$ whose spatial motion is the symmetric stable
process of index $\alpha \in (0,2/3)$, we prove that there exist exceptional
times at which the support is compact and nonempty. Moreover, we show that the
set of exceptional times is dense with full Hausdorff dimension. Besides, we
prove that near extinction, the support of the superprocess is concentrated
arbitrarily close to the distinction point, thus upgrading the corresponding
results in Tribe (1992) from $\alpha \in (0,1/2)$ to $\alpha \in (0,2/3)$, and
we further show that the set of such exceptional times also admits a full
Hausdorff dimension.
