  We present the spectral analysis of three-dimensional dynamics of an elastic
filament in a shear flow of a viscous fluid at a low Reynolds number in the
absence of Brownian motion. The elastica model is used. The fiber initially is
almost straight at an arbitrary orientation, with small perpendicular
perturbations in the shear plane and out-of-plane. To analyze the stability of
both perturbations, equations for the eigenvalues and eigenfunctions are
derived and solved by the Chebyshev spectral collocation method. It is shown
that their crucial features are the same as in the case of the two-dimensional
elastica dynamics in shear flow [Becker and Shelley, Phys. Rev. Lett. 2001] and
the three-dimensional elastica dynamics in the compressional flow [Chakrabarti
et al., Nat. Phys., 2020]. We find a similar dependence of the buckled shapes
on the ratio of bending to hydrodynamic forces as in the simulations for
elastic fibers of a nonzero thickness [Slowicka et al., New J. Phys., 2022].
