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There is a trend towards miniaturization of silicon photonic circuits due to superior performance and small cost. Design rules that must be imposed on the geometry of optical waveguides to make them behave as polarization-independent and single-mode devices are well known for waveguides with relatively large cross sections and for some small cross-sectional rib waveguides with vertical sidewalls and an air top cladding. The influence of the top oxide cover on waveguide birefringence was analyzed recently, but only for relatively large cross-sectional waveguides. This paper reports simulations for both single-mode and polarization-independent behavior for small cross-sectional waveguides with variable rib width, etch depth, top oxide cover thickness, and side-wall angle. The results show that the stress-induced effects must be taken into account to satisfy both requirements. Design rules to maintain birefringence-free operation and to satisfy single-mode behavior for small rib silicon-on-insulator (SOI) waveguides are presented.