  The Euler equations on a three-dimensional periodic domain have a family of
shear flow steady states. We show that the linearised system around these
steady states decomposes into subsystems equivalent to the linearisation of
shear flows in a two-dimensional periodic domain. To do so, we derive a
formulation of the dynamics of the vorticity Fourier modes on a periodic domain
and linearise around the shear flows. The linearised system has a decomposition
analogous to the two-dimensional problem, which can be significantly
simplified. By appealing to previous results it is shown that some subset of
the shear flows are spectrally stable, and another subset are spectrally
unstable. For a dense set of parameter values the linearised operator has a
nilpotent part, leading to linear instability. This is connected to the
nonnormality of the linearised dynamics and the transition to turbulence.
Finally we show that all shear flows in the family considered (even the
linearly stable ones) are parametrically unstable.
