Based on the generalized Mie theory, refractive index sensing characteristics of single silver nanoparticle respectively illuminated by tightly focused linearly-polarized and radially-polarized light beams are investigated. The spectra for localized surface plasmon resonances (LSPR) under different dielectric environments demonstrate that distinct dipolar and quadrupolar resonances can be always observed for the case of radial polarization, while there is only strong dipolar resonance for the case of linear polarization. The dipolar mode has a higher sensitivity. However, the figure of merit (FOM) for the quadrupolar resonance is much higher than that for the dipolar resonance, because the quadrupolar resonance has a narrower width.