We extend the signature-based primal and dual solutions to the optimal stopping problem recently introduced in [Bayer et al.: Primal and dual optimal stopping with signatures, to appear in Finance & Stochastics 2025], by integrating deep-signature and signature-kernel learning methodologies. These approaches are designed for non-Markovian frameworks, in particular enabling the pricing of American options under rough volatility. We demonstrate and compare the performance within the popular rough Heston and rough Bergomi models.