The scramjet engine is a key propulsion system for hypersonic vehicles, leveraging supersonic airflow to achieve high specific impulse, making it a promising technology for aerospace applications. Understanding and controlling the complex interactions between fuel injection, turbulent combustion, and aerodynamic effects of compressible flows are crucial for ensuring stable combustion in scramjet engines. However, identifying stable modes in scramjet combustors is often challenging due to limited experimental measurement means and extremely complex spatiotemporal evolution of supersonic turbulent combustion. This work introduces an innovative deep learning framework that combines dimensionality reduction via the Residual Convolutional Neural Network-beta-Variational Autoencoder (Res-CNN-beta-VAE) model with unsupervised clustering (K-means) to identify and analyze dynamical combustion modes in a supersonic combustor. By mapping high-dimensional data of combustion snapshots to a reduced three-dimensional latent space, the Res-CNN-beta-VAE model captures the essential temporal and spatial features of flame behaviors and enables the observation of transitions between combustion states. By analyzing the standard deviation of latent variable trajectories, we introduce a novel method for objectively distinguishing between dynamic transitions, which provides a scalable and expert-independent alternative to traditional classification methods. Besides, the unsupervised K-means clustering approach effectively identifies the complex interplay between the cavity and the jet-wake stabilization mechanisms, offering new insights into the system's behavior across different gas-to-liquid mass flow ratios (GLRs).