  We analyze the frequency evolution of millisecond oscillations observed
during type I X-ray bursts with the Rossi X-ray Timing Explorer in order to
establish the stability of the mechanism underlying the oscillations. Our
sample contains 68 pulse trains detected in a search of 159 bursts from 8
accreting neutron stars. As a first step, we confirm that the oscillations
usually drift upward in frequency by about 1% toward an apparent saturation
frequency. Previously noted anomalies, such as drifts toward lower frequencies
as the oscillations disappear (``spin-down'' episodes) and instances of two
signals present simultaneously at frequencies separated by a few Hz, occur in
5% of oscillations. Having verified the generally accepted description of burst
oscillations, we proceed to study the coherence of the oscillations during
individual bursts, and the dispersion in the asymptotic frequencies in bursts
observed over five years. On short time scales, we find that 30% of the
oscillation trains do not appear to evolve smoothly in phase. This suggests
either that two signals are present simultaneously with a frequency difference
too small to resolve (< 1 Hz), that the frequency evolution is discontinuous,
or that discrete phase jumps occur. On time scales of years, the maximum
frequencies of the oscillations exhibit fractional dispersions of < 0.004. In
the case of 4U 1636-536, this dispersion is uncorrelated with the known orbital
phase, which indicates that a mechanism besides orbital Doppler shifts prevents
the oscillations from appearing perfectly stable. We interpret these results
under the assumption that the oscillations originate from anisotropies in the
emission from the surfaces of these rotating neutron stars. (Abridged)
