We consider electromagnetic fields having an angular momentum density in a locally non-rotating reference frame in Schwarzschild, Kerr, and Kerr-Newman spacetimes. The state of motion of such fields is assessed with two families of observers, the locally non-rotating ones and those of vanishing Poynting vector comoving with the fields. The examples of relevance in astrophysics and general relativity, such as pure rotating dipolar-like magnetic fields and the electromagnetic field of the Kerr-Newman solution. For the latter example, using Carter observers, we find that the electromagnetic energy is comoving with the gravitational super-energy, which confirms and extends results reported by other authors.