Summary: The present study addresses the classical problem of the dynamics and stability of a cluster of \(N\)-point vortices of equal strength arranged in a polygonal configuration (‘\(N\)-vortex polygons’). In unbounded domains, such \(N\)-vortex polygons are unconditionally stable for \(N\leqslant 7\). Confinement in a circular domain tightens the stability conditions to \(N\leqslant 6\) and a maximum polygon size relative to the domain radius. This work expands on existing studies on stability and integrability by a first giving an exploratory spectral analysis of the dynamics of \(N\) vortex polygons in circular domains. Key to this is that the spectral signature of the time evolution of vortex positions reflects their qualitative behaviour. Expressing vortex motion by a generic evolution operator (the so-called Koopman operator) provides a rigorous framework for such spectral analyses. This paves the way to further differentiation and classification of point-vortex behaviour beyond stability and integrability. The concept of Koopman-based spectral analysis is demonstrated for \(N\)-vortex polygons. This reveals that conditional stability can be seen as a local form of integrability and confirms an important generic link between spectrum and dynamics: discrete spectra imply regular (quasi-periodic) motion; continuous (sub-)spectra imply chaotic motion. Moreover, this exposes rich nonlinear dynamics as intermittency between regular and chaotic motion and quasi-coherent structures formed by chaotic vortices.