We discuss recent theta- and z-pinch experiments performed to investigate the spatially periodic structure of a heavy-current plasma. A heavy-current plasma seeks to form a periodic structure whose basic elements are the E filaments (oriented along the electric field) and the H filaments (oriented perpendicular to the electric field). By means of Rogowski belts and magnetic probes we measured the distributions of plasma currents and magnetic fields. Almost all the plasma current was concentrated in the E filaments that are the elementary z pinches, whereas relatively very small currents flowed along the H filaments. The initial magnetic field stimulated plasma breakup into filaments that were parallel to the field lines. The E- and H-filament systems had a certain stability and they affected the discharge dynamics in a definite manner. Related to them in particular was the excitation of a "tongue" plasma instability in theta pinches.We studied qualitatively several mechanisms that are possible in forming E- and H-filament systems and are based on plasma drift motion in sheared electric and magnetic fields.