In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an ℓp result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillatory component, via a simple method of descent argument. Second, we note an ℓ2 bound for quasi-translation invariant discrete twisted Radon transforms. Finally, we extend an existing ℓ2 bound for a closely related non-translation invariant discrete oscillatory integral operator with singular kernel to an ℓp bound for all 1 < p < 1∞. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function. Copyright © 2010 International Press.