  Various new physics models predict a light CP-odd Higgs boson (labeled as
$a$) and open up new decay modes for Z-boson, such as $Z \to \bar{f} f a$,
$Z\to a\gamma$ and $Z\to aaa$, which could be explored at the GigaZ option of
the ILC. In this work we investigate these rare decays in several new physics
models, namely the type-II two Higgs doublet model (type-II 2HDM), the
lepton-specific two Higgs doublet model (L2HDM), the nearly minimal
supersymetric standard model (nMSSM) and the next-to-minimal supersymmetric
standard model (NMSSM). We find that in the parameter space allowed by current
experiments, the branching ratios can reach $10^{-4}$ for $Z \to \bar{f} f a $
($f=b,\tau$), $10^{-9}$ for $Z\to a\gamma$ and $10^{-3}$ for $Z\to aaa$, which
implies that the decays $Z \to \bar{f} f a$ and $Z \to a a a$ may be accessible
at the GigaZ option. Moreover, since different models predict different
patterns of the branching ratios, the measurement of these rare decays at the
GigaZ may be utilized to distinguish the models.
