Textbooks often hold erroneously that the condition∇·H = 0implies that lines and tubes of force are closed. Examples are given in which the lines of force are not only unclosed, but also, the density of lines may be made infinite in a region where the field intensity is finite. If the steady electric current density is everywhere finite, the magnetic field has singularities only at discrete points, lines, or surfaces, whereH = 0. In general, an infinity of lines of force may terminate on these points. Lines of force may have finite length, going from one singularity to another. A systematic analysis is made of the allowed distribution of singular points, and the linear fields in the neighborhoods of these points are classified. Certain types of singularities may appear in regions of finite current density, but cannot occur in regions free of current. Lines of force may form limit cycles.