  The two-fluid and Landau criteria for superfluidity are compared for trapped
Bose gases. While the two-fluid criterion predicts translational superfluidity,
it is suggested, on the basis of the homogeneous Gross-Pitaevski limit, that a
necessary part of Landau's criterion, adequate for non-translationally
invariant systems, d oes not hold for trapped Bose gases in the GP limit. As a
consequence, if the compressibility is detected to be very large (infinite by
experimental standards) the two-fluid criterion is seen to be the relevant one
in case the system is a tranlational superfluid, while the Landau criterion is
the relevant one if translational superfluidity is absent.
