Light scattering by a small spherical particle with a low dissipation rate is discussed based upon the Mie theory. It is shown that if close to the plasmon (polariton) resonance frequencies the radiative damping prevails over dissipative losses, sharp giant resonances with very unusual properties may be observed. In particular, the resonance extinction cross section increases with an increase in the order of the resonance (dipole, quadrupole, etc.); the characteristic values of electric and magnetic near fields for the scattered light are singular in the particle size, while energy circulation in the near field is rather complicated, so that the Poynting vector field includes singular points whose number, types, and positions are very sensitive to fine changes in the incident light frequency. The results may provide new opportunities for a giant, controlled, highly frequency-sensitive enhancement and variation of electromagnetic field at nanoscales.