Plasmonic refractometric nanosensors based on single nanostructures, i.e. spherical, nanorodand bipyramid-shaped gold nanoparticles, are investigated and compared numerically by employing the finite-difference time-domain method. The results show that the plasmonic sensing ability is distributed anisotropically around the nanorod and bipyramid, even for spherical nanoparticles when the illumination light is linearly polarized. To optimize nanosensor performance, some anisotropy in the shape of nanoparticles is required, this latter serving as an intrinsic light polarization filter to suppress the disturbance from localized surface plasmon resonance in other directions. The plasmonic near-field can be engineered by controlling the shape to achieve a concentrated and localized electromagnetic field, in direct relation with the sensing ability. Taking these factors into account, the gold bipyramid nanoconstruct which is easily available in experiment is proposed as an efficient plasmonic sensing platform. The bipyramid presents both highly localized sensitivity and high scattering cross-section, thus avoiding the trade-off during the selection of the widely used nanorod-shaped sensors. The parameters of the bipyramid structure can be optimized by numerical simulation to improve the plasmonic sensing. Our findings permit a deeper understanding of single-nanoparticle-sensor behavior, and the study provides an opportunity to optimize the plasmonic sensor.