The subject of the paper is oscillation of solutions of second-order delta differential equations of the type
on quite general time scales. Here, is an odd positive integer and are positive right-dense continuous functions. The function may be integrable (in the sense of time scale analysis) at or not. Sufficient conditions for the oscillatory character of all non-trivial solutions of (1) are given in both cases. The main theorems are applicable, in particular, to half-linear difference equations and half-linear ordinary differential equations, unifying and extending previous results. At the end of the paper, some examples are discussed to illustrate the range of applicability of the main results.