The T-matrix formalism is used to investigate multipole resonances exited by electromagnetic plane waves in gold nanorods. Two basic models have been studied: the gold cylinders with semispherical ends (s-cylinders) and prolate gold spheroids with the particle diameters 20 and 40 nm and the aspect ratios ranging from 2 to 20. The extinction, absorption, and scattering spectra were calculated for TM and TE incident wave (the orientation angle was varied from 0 to 90o), as well as for randomly oriented particles. Dependent on the particle size and shape, we observed from 2 to 8 multipole resonances in calculated spectra. On the basis of these simulations, we have derived an explicit rule for partial multipole contributions to the plasmon resonances of gold nanorods at a fixed or random orientation. The parity of a given spectral resonance number n coincides with the parity of their multipole contributions l, where l is equal to or greater than n, and the total resonance magnitude is determined by the lowest multipole contribution. We also investigate the dependence of multipole plasmons on the orientation of nanorods with respect to polarized incident light, as well as on the particle size, shape, and dielectric environment. The relative shift of the low multipole resonances (n=2-5) is proportional to the relative increment of the medium refractive index. It is shown that the multipole resonance wavelengths as a function of the aspect ratio divided by the resonance number collapse onto one linear scaling curve. This scaling is explained by using the plasmon standing wave concept introduced by Schider et al. (Phys. Rev. B, 2003, 68, 155427). Although the multipole scaling reflects some universal qualitative properties of the standing wave physics, the quantitative scaling parameters depend on the particle size and morphology.