Extracting governing equations from measurement data is a significant challenge in science and engineering. Despite abundant data, creating accurate, reliable models for analysis and prediction remains complex, requiring careful design, validation, and domain-specific knowledge. This study focuses on identifying nonlinear nonsmooth dynamics from data. While the Sparse Identification of Nonlinear Dynamics (SINDy[1]) algorithm works well for smooth dynamics, it struggles with nonsmooth systems. We propose a novel method tailored for nonlinear nonsmooth dynamics. We tested two hypotheses: (1) dimensional reduction is unnecessary, and (2) state variables involved in nonsmooth nonlinearities are observable. We explored two scenarios: one where nonsmooth components are unidentifiable (current state of the art) and another where they are identifiable as monic polynomial functions. For the latter, we combined a line search algorithm with rupture detection to identify subdifferentials and used a Bayesian spike-and-slab approach to handle noisy data. Future work should address the uniqueness of differential equations derived from input data and generalize findings to systems with $n$ degrees of freedom. While we focused on simple monic polynomial functions, broader methods are needed for general nonsmooth functions. We propose integrating the SINDy framework with neural networks, using basis functions as activation functions to enhance nonsmooth component identification. This approach leverages neural networks' flexibility and learning capabilities. Multiplicative neural networks (MNNs) (see [2]) are introduced as a promising tool. Unlike traditional networks, MNNs use multiplicative interactions between inputs, enabling them to capture complex feature dependencies. This architecture improves interpretability and performance in modeling intricate dynamics, making it well-suited for identifying nonsmooth systems.