  Several string theories related to QCD in two dimensions are studied. For
each of these theories the large $N$ free energy on a (target) sphere of area
$A$ is calculated. By considering theories with different subsets of the
geometrical structures involved in the full QCD${}_2$ string theory, the
different contributions of these structures to the string free energy are
calculated using both analytic and numerical methods. The equivalence between
the leading terms in the $SU(N)$ and $U(N)$ free energies is simply
demonstrated from the string formulation. It is shown that when $\Omega$-points
are removed from the theory, the free energy is convergent for small and large
values of $A$ but divergent in an intermediate range. Numerical results
indicate that the free energy for the full QCD${}_2$ string fails to converge
at the Douglas-Kazakov phase transition point. Similar results for a single
chiral sector of the theory, such as has recently been studied by Cordes,
Moore, and Ramgoolam, indicate that there are three distinct phases in that
theory. These results indicate that from the point of view of the strong
coupling phase, the phase transition in the full QCD${}_2$ string arises from
the entropy of branch-point singularities.
