The reliability of machine unlearning critically depends on the integrity and fidelity of the data to be forgotten. In practice, forget sets may deviate from the original training distribution desired for unlearning due to noise, rewriting, or deliberate modifications such as watermarking. These variations raise important questions, e.g., how do different types of data shift, from random perturbations to structured transformations, affect unlearning efficacy? When is forgetting robust to imperfect data, and when does performance degrade? Can perturbations, rather than being mere obstacles, be exploited to improve forgetting? This chapter synthesizes research on data perturbation–centric machine unlearning, examining how perturbations at both test time and training time influence forgetting efficacy, model utility, and robustness. We begin with test-time data shifts, showing that while perturbations such as Gaussian noise, elastic transformations, and adversarial attacks degrade model utility, they often leave unlearning performance unaffected, or even enhanced, by amplifying forgetting signals. Extending to training-time shifts, we analyze incomplete, rewritten, and watermarked forget sets, finding that unlearning effectiveness depends primarily on the semantic fidelity rather than the surface form of forget data: as long as core concepts remain intact, forgetting proceeds reliably despite lexical or structural alterations. Finally, we demonstrate that watermarking can be transformed from a passive perturbation into an active unlearning signal through frameworks such as Water4MU, which deliberately embed semantically consistent yet separable patterns into forget data to improve forgetting accuracy, robustness against membership inference, and even safety in generative models, all while preserving model utility.