We present a complete derivation of the semiclassical limit of the
coherent-state propagator in one dimension, starting from path integrals
in phase space. We show that the arbitrariness in the path integral representation,
which follows from the overcompleteness of the coherent states, results
in many different semiclassical limits. We explicitly derive two possible
semiclassical formulae for the propagator, we suggest a third one, and
we discuss their relationships. We also derive an initial-value representation
for the semiclassical propagator, based on an initial Gaussian wavepacket.
It turns out to be related to, but different from, Heller s thawed Gaussian
approximation. It is very different from the Herman Kluk formula, which
is not a correct semiclassical limit. We point out errors in two derivations
of the latter. Finally we show how the semiclassical coherent-state propagators
lead to WKB-type quantization rules and to approximations for the Husimi
distributions of stationary states.