Topological singularities are fundamental organizing principles in wave physics, yet whether nonlinearity can intrinsically generate new singular structures remains largely unexplored. Here, we show that even a minimal nonlinear system—consisting of a single Kerr-nonlinear cavity coupled to a linear cavity—supports a rich hierarchy of topological singularities with no counterpart in linear two-mode systems. By analyzing the full nonlinear spectrum, we identify multiple classes of degeneracies, including higher-order exceptional-like singularities, nondefective degeneracies, and unconventional second-order singularities exhibiting anomalous scaling behavior. To rigorously define their topology, we introduce an isospectral embedding that maps the nonlinear eigenproblem to a higher-dimensional linear Hamiltonian, enabling well-defined Berry-phase and phase-rigidity diagnostics. This framework reveals that commonly studied limits, such as the zero-coupling case, correspond to special cross-sections of a broader nonlinear singularity landscape. Our results establish nonlinearity as a fundamental source of topological singularities and provide a general route toward higher-order and unconventional topology beyond linear paradigms.

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• Nonlinear optics
• Topological effects in photonic systems
• Non-Hermitian systems
• Nonlinear waves
• PT-symmetry
• Topology