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The characteristic interactions of discrete modes supported by planar isotropic and anisotropic dielectric slab waveguides are analyzed using singularity and critical point theory, leading to a rigorous and complete explanation of all modal interactions. Complex frequency-plane singularities associated with modes on isotropic waveguides are identified and discussed, and the absence of mode coupling is proven. For an anisotropic planar waveguide having an arbitrarily oriented optical axis, it is shown that mode coupling is controlled by the presence of an isolated Morse critical point (MCP) accompanied by a pair of complex-conjugate frequency-plane branch points. The interaction of space-wave leaky modes on a grounded anisotropic slab is studied by investigating the evolution of complex frequency-plane branch point singularities as the orientation of the optical axis varies. The general theory is presented, and numerical results are provided for some specific waveguides.