Multimode waveguides on lithium niobate-on-insulator (LNOI) and silicon-on-insulator (SOI) platforms are numerically investigated in buried, rib, and strip configurations. Performance of waveguides is compared in terms of waveguide cross-sectional area, dispersion, mode hybridization, and power confinement for both quasi-transverse electric and quasi-transverse magnetic modes. Tall waveguides with single mode in the horizontal direction, supporting higher order modes in the vertical direction, are analyzed. Also, wide waveguides with single mode in the vertical direction, supporting higher order modes in the horizontal direction, are studied. Designs that overcome mode hybridization are proposed, which are well suited for applications such as optical interconnects. LNOI waveguides were found to exhibit lower dispersion in all the configurations, with power confinement and physical dimensions comparable to those of SOI waveguides. The results are instrumental in design optimization of multimode components for on-chip mode-division multiplexing schemes and multiparameter sensing applications. Electro-optic effect is also illustrated in a buried multimode LNOI waveguide that is largely useful in modulation and switching applications.