In this paper, spectral finite elements (SFEs) are developed for wave propagation analysis of isotropic curved beams using three different beam models: (1) the refined third-order shear deformation theory (TOT), (2) the first-order shear deformation theory (FSDT), and (3) the classical shell theory (CST). The formulation is validated by comparing the results for the wavenumber dispersion relations and natural frequencies with the published results based on the FSDT. The numerical study reveals that even for a very thin curved beam with radius-to-thickness ratio of 1000, the wavenumbers predicted by the CST at high frequencies show significant deviation from those of the shear deformable theories, FSDT and TOT. The FSDT results for the wavenumber of the flexural displacement mode differ significantly from the TOT results at high frequencies even for thin beams. The deviation increases and occurs at lower frequencies with the decrease in the radius-to-thickness ratio. The results for wave propagation response show that the CST yields highly erroneous response for flexural mode wave propagation even for thin beams and at a relatively low frequency of 20 kHz. The FSDT results too differ by unacceptably high margin from the TOT results for flexural wave response of thin beams at frequencies greater than 100 kHz, which are typically used for structural health monitoring (SHM) applications. For thick beams, FSDT results for the tangential wave response also show large deviation from the TOT results.