  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.
