Deep learning systems achieve remarkable empirical performance, yet the stability of the training process itself remains poorly understood. Training unfolds as a high-dimensional dynamical system in which small perturbations to optimization, data, parameters, or learning signals can induce abrupt and irreversible collapse, undermining reproducibility and scalability. We propose a unified dynamical perspective that characterizes training stability as an intrinsic property of learning systems, organized along four interacting dimensions: optimization, environmental/data, parametric, and learning-signal stability. We operationalize this perspective through controlled perturbation auditing of training trajectories, probing how learning dynamics respond to structured disturbances without modifying learning algorithms. Across reinforcement learning and large language model training, we identify three recurring regularities: high final performance is frequently decoupled from training stability; controlled stochasticity consistently buffers learning dynamics across paradigms; and deviations in low-dimensional latent meta-states systematically precede observable performance collapse. Together, these findings establish training stability as a measurable and comparable dynamical property of learning systems, providing a descriptive foundation for studying learning dynamics beyond final performance outcomes.