The plasmon resonances of two-dimensional (2D) metallic particles with typical sizes of 15-50 nm are numerically investigated. Extremely large electromagnetic fields enhancements occur near the surface of the particle at resonance frequencies. These emergent localized surface plasmons are governed by both the dielectric function of the metallic object, its geometry and the illumination direction. The analysis of such spherical and triangle-shaped nanoparticles is performed using the semi-analytical multiple multipole (MMP) method, which provides complex eigenvalue estimations as well as proper error measures. While controlling error measures highly-accurate computations are carried out especially for coupled-particle configurations with different inter-particle distances and under various illumination conditions.