  In this paper we find all solvable subgroups of Diff^omega(S^1) and classify
their actions. We also investigate the C^r local rigidity of actions of the
solvable Baumslag-Solitar groups on the circle.
  The investigation leads to two novel phenomena in the study of infinite group
actions on compact manifolds. We exhibit a finitely generated group Gamma and a
manifold M such that:
  * Gamma has exactly countably infinitely many effective real-analytic actions
on M, up to conjugacy in Diff^omega(M);
  * every effective, real analytic action of Gamma on M is C^r locally rigid,
for some r>=3, and for every such r, there are infinitely many nonconjugate,
effective real-analytic actions of Gamma on M that are C^r locally rigid, but
not C^(r-1) locally rigid.
