  Although it is conjectured that a phase transition from hadronic to
deconfined quark matter in the ultrahigh-density environment of Neutron Stars
(NS), the nature of phase transition remains an unresolved mystery.
Furthermore, recent efforts reveal that the finite surface tension effects can
lead to a mixed phase with different geometric shapes (so-called "pasta"
phases), leading to a smooth phase transition from hadronic to quark matter in
the NS interior. Depending on whether there is a strong or a pasta-induced
smooth first-order phase transition, one may expect a third family of stable,
compact stars or "twin stars" to appear, with the same mass but different radii
compared to NSs. The possibility of identifying twin stars using astrophysical
observations has been a subject of interest. This study investigates the
potential of probing the nature of the hadron-quark phase transition through
future gravitational wave (GW) detections from fundamental ($f$-) mode
oscillations in Neutron Stars. Using a newly developed model that parametrizes
the hadron-quark phase transition with ``pasta phases," we calculate $f$-mode
characteristics within a full general relativistic framework. We then use
Universal Relations in GW asteroseismology to derive stellar properties from
the detected mode parameters. Our findings suggest that detecting GWs from
$f$-modes with third-generation GW detectors offers a promising scenario for
the existence of twin stars. However, we also estimate various uncertainties in
determining the mode parameters and conclude that these uncertainties make it
more challenging to identify the nature of the hadron-quark phase transition.
