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Let $f$ be a regular real-valued non-constant symbol defined on the onedimensional torus ${\mathbb T}$. Denote respectively by $\kappa$ and $T$, itsset of critical points and the associated Toeplitz matrix on $l^2({\mathbbN})$. If $V$ is a suitable compact perturbation, we prove that the operator$T+V$ has no singular continuous spectrum and only finite point spectrum awayfrom the set of thresholds $f(\kappa)$. We also obtain some propagationestimates and apply these results to concrete examples.

Spectral stability for compact perturbations of Toeplitz matrices