The authors consider the operator \(L\) generated in \(\ell^2({\mathbb Z})\) by the difference expression \((\ell y)_n=a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}\), \(n\in{\mathbb Z}\), where \(\{a_n\}_{n\in{\mathbb Z}}\) and \(\{b_n\}_{n\in{\mathbb Z}}\) are complex sequences. The spectrum, the spectral singularities, and the properties of the principal vectors corresponding to the spectral singularities of \(L\) are investigated. The authors also study similar problems for the discrete Dirac operator generated in \(\ell({\mathbb Z,\mathbb C}^2)\) by the system of the difference expression \[ \begin{pmatrix} \Delta y_n^{(2)}+p_ny_n^{(1)}\cr -\Delta y_{n-1}^{(1)}+q_ny_n^{(2)} \end{pmatrix}, \] \(n\in{\mathbb Z}\), where \(\{p_n\}_{n\in{\mathbb Z}}\) and \(\{q_n\}_{n\in{\mathbb Z}}\) are complex sequences.