We observe that a Polish group $G$ is amenable if and only if every
continuous action of $G$ on the Hilbert cube admits an invariant probability
measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that
actions on the Cantor space can be used to detect amenability and extreme
amenability of Polish non-archimedean groups as well as amenability at infinity
of discrete countable groups. As corollary, the latter property can also be
tested by actions on the Hilbert cube. These results generalise a criterion due
to Giordano and de la Harpe.