  Self-similar solutions provide good descriptions for the gravitational
collapse of spherical clouds or stars when the gas obeys a polytropic equation
of state, $p=K\rho^\gamma$ (with $\gamma\le 4/3$). We study the behaviors of
nonradial perturbations in the similarity solutions of Larson, Penston and
Yahil, which describe the evolution of the collapsing cloud prior to core
formation. Our global stability analysis reveals the existence of unstable
bar-modes ($l=2$) when $\gamma\le 1.09$. In particular, for the collapse of
isothermal spheres, which applies to the early stages of star formation, the
$l=2$ density perturbation relative to the background, $\delta\rho({\bf
r},t)/\rho(r,t)$, increases as $(t_0-t)^{-0.352}\propto \rho_c(t)^{0.176}$,
where $t_0$ denotes the epoch of core formation, and $\rho_c(t)$ is the cloud
central density. Thus, the isothermal cloud tends to evolve into an ellipsoidal
shape (prolate bar or oblate disk, depending on initial conditions) as the
collapse proceeds. In the context of Type II supernovae, core collapse is
described by the $\gamma\simeq 1.3$ equation of state, and our analysis
indicates that there is no growing mode (with density perturbation) in the
collapsing core before the proto-neutron star forms, although nonradial
perturbations can grow during the subsequent accretion of the outer core and
envelope onto the neutron star. We also carry out a global stability analysis
for the self-similar expansion-wave solution found by Shu, which describes the
post-collapse accretion (``inside-out'' collapse) of isothermal gas onto a
protostar. We show that this solution is unstable to perturbations of all
$l$'s, although the growth rates are unknown.
