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Robust measurements of multiple black hole vibrational modes provide a uniqueopportunity to characterise gravity in extreme curvature and dynamical regimes,to better investigate the nature of compact objects and search for signs of newphysics. We use a numerically-tuned quasicircular non-precessing ringdownmodel, $\texttt{TEOBPM}$, and the $\texttt{pyRing}$ analysis infrastructure toperform a time-domain spectroscopic analysis of the third catalog of transientgravitational-wave signals, GWTC-3, searching for higher angular modes. The$\texttt{TEOBPM}$ model effectively includes non-linearities in the earlypost-merger signal portion, and carries information about the progenitorsparameters through time-dependent excitation amplitudes of the black holequasinormal modes. Such a strategy allows us to accurately model the fullpost-merger emission, recovering higher signal-to-noise ratios compared totemplates based on more agnostic superpositions of damped-sinusoids, at theexpense of obtaining less agnostic results. We find weak evidence for the$(l,m)=(3,3)$ [$(l,m)=(2,1)$] mode in several events, with the largest Bayesfactor in favour of this mode being $B \simeq 2.6$ [$B \simeq 1.2$] within thesupport of the peak time distribution, of which we crucially characterise theimpact on the detection. For GW190521, we observe $B \simeq 5.1$ , but only fortimes outside the peak time support reconstructed using the highly accurate$\texttt{NRSur7dq4}$ model, indicating significant systematics affecting suchputative detection. Allowing for deviations from general relativity under theassumption of the presence of two modes, we find tentative support for the Kerr"final state conjecture". Our work showcases a systematic methodology torobustly identify and characterise higher angular modes in ringdown-onlysignals [abridged]

Searching for ringdown higher modes with a numerical relativity-informed post-merger model