  Supersolidity is deeply connected with the emergence of Goldstone modes,
reflecting the spontaneous breaking of both phase and translational symmetry.
Here, we propose accessible signatures of these modes in harmonically trapped
spin-orbit-coupled Bose-Einstein condensates, where supersolidity appears in
the form of stripes. By suddenly changing the trapping frequency, an axial
breathing oscillation is generated, whose behavior changes drastically at the
critical Raman coupling. Above the transition, a single mode of hybridized
density and spin nature is excited, while below it, we predict a beating effect
signaling the excitation of a Goldstone spin-dipole mode. We further provide
evidence for the Goldstone mode associated with the translational motion of
stripes. Our results open up new perspectives for probing supersolid properties
in experimentally relevant configurations with both symmetric as well as highly
asymmetric intraspecies interactions.
