Machine learning algorithms are vulnerable to data poisoning attacks. Prior taxonomies that focus on specific scenarios, e.g., indiscriminate or targeted, have enabled defenses for the corresponding subset of known attacks. Yet, this introduces an inevitable arms race between adversaries and defenders. In this work, we study the feasibility of an attack-agnostic defense relying on artifacts that are common to all poisoning attacks. Specifically, we focus on a common element between all attacks: they modify gradients computed to train the model. We identify two main artifacts of gradients computed in the presence of poison: (1) their $\ell_2$ norms have significantly higher magnitudes than those of clean gradients, and (2) their orientation differs from clean gradients. Based on these observations, we propose the prerequisite for a generic poisoning defense: it must bound gradient magnitudes and minimize differences in orientation. We call this gradient shaping. As an exemplar tool to evaluate the feasibility of gradient shaping, we use differentially private stochastic gradient descent (DP-SGD), which clips and perturbs individual gradients during training to obtain privacy guarantees. We find that DP-SGD, even in configurations that do not result in meaningful privacy guarantees, increases the model's robustness to indiscriminate attacks. It also mitigates worst-case targeted attacks and increases the adversary's cost in multi-poison scenarios. The only attack we find DP-SGD to be ineffective against is a strong, yet unrealistic, indiscriminate attack. Our results suggest that, while we currently lack a generic poisoning defense, gradient shaping is a promising direction for future research.