  We study the spectral stability of the nonlinear Dirac operator in dimension
$1+1$, restricting our attention to nonlinearities of the form
$f(\langle\psi,\beta \psi\rangle_{\mathbb{C}^2}) \beta$. We obtain bounds on
eigenvalues for the linearized operator around standing wave solutions of the
form $e^{-i\omega t} \phi_0$. For the case of power nonlinearities $f(s)= s
|s|^{p-1}$, $p>0$, we obtain a range of frequencies $\omega$ such that the
linearized operator has no unstable eigenvalues on the axes of the complex
plane. As a crucial part of the proofs, we obtain a detailed description of the
spectra of the self-adjoint blocks in the linearized operator. In particular,
we show that the condition $\langle\phi_0,\beta \phi_0\rangle_{\mathbb{C}^2} >
0$ characterizes groundstates analogously to the Schr\"odinger case.
