  Quasi-normal modes of NSs are an exciting prospect for analyzing the internal
composition of NSs and studying matter at high densities. In this work, we
focus on studying the $f$- and $p$- quadrupolar oscillation modes, which couple
with gravitational waves. We construct two different EOS ensembles, one without
and one with a first-order phase transition, and examine how $f$- and $p$-modes
might help us differentiate them. We find ensemble specific exclusion regions
in the $65\%$ and $95\%$ confidence contours of the frequency-damping time
relations. The exclusion regions become more prominent for the higher-order
oscillation modes. However, these modes have higher frequencies, which are
beyond the detection capabilities of present gravitational wave detectors. The
quasi-universal relations of dimensionless quantities prove to be ineffective
in differentiating the EOS ensembles, as they obscure the details of the EOS.
