In three-dimensional (3D) medical ultrasound imaging with two-dimensional (2D) arrays, sparse 2D arrays have been studied to reduce the number of active channels. Among them, sparse 2D arrays with regular or uniform arrangements of elements have advantages of low side lobe energy and uniform field responses over the entire field of view. This paper presents two uniform sparse array models: sparse rectangular arrays (SRAs) on a rectangular grid and sparse spiral arrays (SSAs) on a sunflower grid. Both arrays can be easily implemented on the commercially available or the custom-made arrays. To suppress the overall grating lobe levels, the transmit (Tx) and receive (Rx) array pairs of both the array models are designed not to have grating lobes at the same locations in the Tx/Rx beam patterns, for which the theoretical design rules are also proposed. Computer simulation results indicate that the proposed array pairs for both the SRAs and the SSAs achieve peak grating lobe levels below -40 dB using about a quarter of the number of elements in the dense rectangular array while maintaining similar beam widths to that of the dense array pair.