  We present a general approach to the analysis of gauge stability of 3+1
formulations of General Relativity (GR). Evolution of coordinate perturbations
and the corresponding perturbations of lapse and shift can be described by a
system of eight quasi-linear partial differential equations. Stability with
respect to gauge perturbations depends on a choice of gauge and a background
metric, but it does not depend on a particular form of a 3+1 system if its
constrained solutions are equivalent to those of the Einstein equations.
Stability of a number of known gauges is investigated in the limit of
short-wavelength perturbations. All fixed gauges except a synchronous gauge are
found to be ill-posed. A maximal slicing gauge and its parabolic extension are
shown to be ill-posed as well. A necessary condition is derived for
well-posedness of metric-dependent algebraic gauges. Well-posed
metric-dependent gauges are found, however, to be generally unstable. Both
instability and ill-posedness are associated with perturbations of physical
accelerations of reference frames.
