Efficient zero location tests for the delta-operator-based polynomials have been a recent topic of interest. These tests not only provide the needed efficient tests for the delta-operator formulated systems, control, and signal processing problems, which are numerically better behaved than the traditional shift-operator based tests for fast sampling, they also provide missing links between classical tests in the z-domain and the s-domain such as the Schur-Cohn test (SCT) and the Routh test (RT). These will also be accomplished in this paper, but for the singular cases. Specifically, the recently developed singular case I Schur-Cohn test by Pal and Kailath (1994) is modified and transformed into the delta-domain. As the sampling interval vanishes this new algorithm converges to an s-domain singular case I test associated with the recently developed s-domain SCT. Singular case II is also considered and the difference between the traditional approaches in the z-domain and the s-domain reconciled. Numerical examples are given to illustrate better numerical properties of the new test over Pal-Kailath recursion for fast sampling.