Similar to electromagnetic (EM) phenomena, described by Maxwell equations, physics of magnetoelectric (ME) phenomena deals with the fundamental problems of the relationship between electric and magnetic fields. The different nature of these two notions is especially evident in dynamic regimes. Analyzing the EM phenomena inside the ME material, the question arises: What kind of the near fields, originated from a sample of such a material, can be measured? Observation of the ME states requires an experimental technique characterized by a violation of spatial and temporal inversion symmetries in a subwavelength region. This presumes the existence of specific near fields. Recently, such field structures, called ME fields, were found as the near fields of a quasi-2D subwavelength-size ferrite disk with magnetic-dipolar-mode (MDM) oscillations. The key physical characteristics that determine the configurations of the ME near fields are the spin and orbital angular momenta of the quantum states of the MDM spectra. This leads to the appearance of subwavelength power-flow vortices. By virtue of unique topology, the ME quantum fluctuations in vacuum are different from virtual EM photons. While preserving the ME properties, one observes strong enhancing the near-field intensity. The main purpose of this chapter is to review and analyze the studies of the ME fields. We consider the near-field topological singularities originated from the MDM ferrite-disk particle. These topological features can be transmitted to various types of nonmagnetic material structures.