  We explore self-similar dynamical processes in a spherical isothermal
self-gravitational fluid with an emphasis on shocks and outline astrophysical
applications of such shock solutions. The previous similarity shock solutions
of Tsai & Hsu and of Shu et al. may be classified into two types: Class I
solutions with downstream being free-fall collapses and Class II solutions with
downstream being Larson-Penston (LP) type solutions. By the analyses of Lou &
Shen and Shen & Lou, we further construct similarity shock solutions in the
`semi-complete space'. These general shock solutions can accommodate and model
dynamical processes of radial outflows (wind), inflows (accretion or
contraction), subsonic oscillations, and free-fall core collapses all with
shocks in various settings such as star-forming molecular clouds, `champagne
flows' in H{\sevenrm II} regions around luminous massive OB stars or
surrounding quasars, dynamical connection between the asymptotic giant branch
phase to the proto-planetary nebula phase with a central hot white dwarf as
well as accretion shocks around compact objects such as white dwarfs, neutron
stars, and supermassive black holes. By a systematic exploration, we are able
to construct families of infinitely many discrete Class I and Class II
solutions matching asymptotically with a static outer envelope of singular
isothermal sphere; the shock solutions of Tsai & Hsu form special subsets.
These similarity shocks travel at either subsonic or supersonic constant
speeds. We also construct twin shocks as well as an `isothermal shock'
separating two fluid regions of two different yet constant temperatures.
