For an arbitrary Hermitian period-T Jacobi operator, we assume a per- turbation by a Wigner{von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solutions. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi opera- tor, which are stable with respect to an l1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.