  We examine the spectral stability and instability of periodic traveling waves
for regularized long-wave models. Examples include the regularized Boussinesq,
Benney--Luke, and Benjamin--Bona--Mahony equations. Of particular interest is a
striking new instability phenomenon -- spectrum off the imaginary axis
extending into infinity. The spectrum of the linearized operator of the
generalized Korteweg--de Vries equation, for instance, lies along the imaginary
axis outside a bounded set. The spectrum for a regularized long-wave model, by
contrast, can vary markedly with the parameters of the periodic traveling
waves. We carry out asymptotic spectral analysis to short wavelength
perturbations, distinguishing whether the spectrum tends to infinity along the
imaginary axis or some curve whose real part is nonzero. We conduct numerical
experiments to corroborate our analytical findings.
