  This paper studies both existence and spectral stability properties of
bounded spatially periodic traveling wave solutions to a large class of scalar
viscous balance laws in one space dimension with a reaction function of
monostable or Fisher-KPP type. Under suitable structural assumptions, it is
shown that this class of equations underlies two families of periodic waves.
The first family consists of small amplitude waves with finite fundamental
period which emerge from a Hopf bifurcation around a critical value of the wave
speed. The second family pertains to arbitrarily large period waves which arise
from a homoclinic bifurcation and tend to a limiting traveling (homoclinic)
pulse when their fundamental period tends to infinity. For both families, it is
shown that the Floquet (continuous) spectrum of the linearization around the
periodic waves intersects the unstable half plane of complex values with
positive real part, a property known as spectral instability. For that purpose,
in the case of small-amplitude waves it is proved that the spectrum of the
linearized operator around the wave can be approximated by that of a constant
coefficient operator around the zero solution and determined by a dispersion
relation which intersects the unstable complex half plane. In the case of large
period waves, we verify that the family satisfies the assumptions of the
seminal result by Gardner (1997, J. Reine Angew. Math. 491, pp. 149-181) of
convergence of periodic spectra in the infinite-period limit to that of the
underlying homoclinic wave, which is unstable. A few examples are discussed.
