We study a range of radio-frequency ion-trap geometries and investigate the effect of integrating dielectric cavity mirrors on their trapping potential using numerical modeling. We compare five different ion-trap geometries with the aim to identify ion-trap and cavity configurations that are best suited for achieving small cavity volumes and thus large ion-photon coupling as required for scalable quantum-information networks. In particular, we investigate the trapping-potential distortions caused by the dielectric material of the cavity mirrors in all three dimensions for different mirror orientations with respect to the trapping electrodes. We also analyze the effect of the mirror material properties such as dielectric constants and surface conductivity, and study the effect of surface charges on the mirrors. As well as perfectly symmetric systems, we also consider traps with optical cavities that are not centrally aligned where we find a spatial displacement of the trap center and asymmetry of the resulting trap only at certain cavity orientations. The best trap-cavity configurations with the smallest trapping-potential distortions are those where the cavities are aligned along the major symmetry axis of the electrode geometries. These cavity configurations also appear to be the most stable with respect to any mirror misalignment. Although we consider particular trap sizes in our study, the presented results can be easily generalized and scaled to different trap dimensions.