Nonlinear eigenvalue problems (NLEVPs) arise in thermoacoustics by considering the temporal evolution of small perturbations in the relevant governing equations. In this work, two solution strategies are compared: (i) a contour-integration-based method that guarantees to provide all eigenvalues in a given domain and (ii) a method that approximates the NLEVP by a rational eigenvalue problem (REVP), which is generally easier to solve. The focus lies on numerical speed, the completeness of the computed spectrum, and the appearance of spurious modes, i.e., modes that are not part of the original spectrum but appear as a result of the approximation. To this end, two prototypical thermoacoustic systems are considered: a single-flame Rijke tube and an annular model combustor. The comparison of both methods is preceded by a detailed analysis of the user-defined input parameters in the contour-integration-based method. Our results show that both methods can resolve all types of considered eigenvalues with sufficient accuracy for applications. However, the recast linear problem is overall faster to solve and allows a priori precision estimates—unlike the contour-integration-based method. Spurious modes as a by-product of the NLEVP approximation are found to play a minor role, and recommendations are given on how to eliminate them.