In thermally anisotropic plasmas or sheet pinches, the zero‐order counterstreaming or directed streaming of like particles gives rise to instability modes that correspond essentially to the formation of local first‐order pinches within the plasma. The marginal stability criteria for all such modes can be interpreted in terms of Bennett's pinch condition. The range of examples discussed includes Weibel's thermally anisotropic hot‐electron plasma instability in null field, which is generalized for hot ions and finite magnetic field, and which is applied to interpenetrating neutral plasma streams, to a spherical electrostatic plasma‐containment scheme, and to the Astron thermonuclear experiment. The familiar ``mirror instability'' is shown to be another example of a ``pinch''‐type mode. Sheet pinches with or without longitudinal magnetic field are generally unstable against tearing into subsidiary linear pinches.