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Optical nanofiber cavity research has mainly focused on the fundamental mode. Here, a Fabry-P\'erot fiber cavity with an optical nanofiber supporting the higher-order modes, TE01, TM01, HE21o, and HE21e, is demonstrated. Using cavity spectroscopy, with mode imaging and analysis, we observe cavity resonances that exhibit complex, inhomogeneous states of polarization with topological features containing Stokes singularities such as C-points, Poincar\'e vortices, and L-lines. In situ tuning of the intracavity birefringence enables the desired profile and polarization of the cavity mode to be obtained. These findings open new research possibilities for cold atom manipulation and multimode cavity quantum electrodynamics using the evanescent fields of higher-order mode optical nanofibers.