  We study the linear stability of localized modes in self-interacting spinor
fields, analyzing the spectrum of the operator corresponding to linearization
at solitary waves. Following the generalization of the Vakhitov--Kolokolov
approach, we show that the bifurcation of real eigenvalues from the origin is
completely characterized by the Vakhitov--Kolokolov condition $dQ/d\omega=0$
and by the vanishing of the energy functional. We give the numerical data on
the linear stability in the generalized Gross--Neveu model and the generalized
massive Thirring model in the charge-subcritical, critical, and supercritical
cases, showing the agreement with the Vakhitov--Kolokolov and the energy
vanishing conditions.
