  Networks of coupled phase oscillators play an important role in the analysis
of emergent collective phenomena. In this article, we introduce generalized
$m$-splay states constituting a special subclass of phase-locked states with
vanishing $m$th order parameter. Such states typically manifest incoherent
dynamics, and they often create high-dimensional families of solutions (splay
manifolds). For a general class of phase oscillator networks, we provide
explicit linear stability conditions for splay states and exemplify our results
with the well-known Kuramoto-Sakaguchi model. Importantly, our stability
conditions are expressed in terms of just a few observables such as the order
parameter or the trace of the Jacobian. As a result, these conditions are
simple and applicable to networks of arbitrary size. We generalize our findings
to phase oscillators with inertia and adaptively coupled phase oscillator
models.
