Recent single-molecule experiments have observed that formation of a plectonemically supercoiled region in a stretched, twisted DNA proceeds via abrupt formation of a small plectonemic ''bubble". A detailed mesoscopic model is presented for the formation of plectonemic domains, including their positional entropy, and the influence of small chiral loops or ''curls'' along the extended DNA. Curls begin to appear just before plectoneme formation, and are more numerous at low salt concentrations ($< 20$ mM univalent ions) and at low forces ($< 0.5$ pN). However, plectonemic domains quickly become far more stable slightly beyond the transition to supercoiling at moderate forces and physiological salt conditions. At the supercoiling transition for shorter DNAs (2 kb), only one supercoiled domain appears, but for longer DNAs at lower forces ($< 0.5$ pN), positional entropy favors formation of more than one plectonemic domain; a similar effect occurs for low salt. Although they are not the prevalent mode of supercoiling, curls are a natural transition state for binding of DNA-loop-trapping enzymes; we show how addition of loop-trapping enzymes can modify the supercoiling transition. The behavior of DNA torque is also discussed, including the effect of the measurement apparatus torque stiffness, which plays a role in determining how large the torque ''overshoot" is at the buckling transition.