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The velocity particle distributions measured in-situ in space plasmas deviate from Maxwellian (thermal) equilibrium, showing enhanced suprathermal tails which are well described by the standard Kappa-distribution (SKD). Despite its successful application, the SKD is frequently disputed due to a series of unphysical implications like diverging velocity moments, preventing a macroscopic description of the plasma. The regularized Kappa-distribution (RKD) has been introduced to overcome these limitations, but the dispersion properties of RKD-plasmas are not explored yet. In the present paper we compute the wavenumber dispersion of the frequency and damping or growth rates for the electromagnetic modes in plasmas characterized by the RKD. This task is accomplished by using the grid-based kinetic dispersion solver LEOPARD developed for arbitrary gyrotropic distributions [P. Astfalk and F. Jenko, J. Geophys. Res. 122, 89 (2017)]. By reproducing previous results obtained for the SKD and Maxwellian, we validate the functionality of the code. Furthermore, we apply the isotropic as well as the anisotropic RKDs to investigate stable electromagnetic electron-cyclotron (EMEC) and ion-cyclotron (EMIC) modes as well as temperature-anisotropy-driven instabilities, both for the case $T_\perp / T_\parallel > 1$ (EMEC and EMIC instabilities) and for the case $T_\perp / T_\parallel < 1$ (proton and electron firehose instabilities), where $\parallel$ and $\perp$ denote directions parallel and perpendicular to the local time-averaged magnetic field. Provided that the cutoff parameter $α$ is small enough, the results show that the RKDs reproduce the dispersion curves of the SKD plasmas at both qualitative and quantitative levels. For higher values, however, physically significant deviation occurs.