Complicated cosmic string loops will fragment until they reach simple, non-intersecting ("stable") configurations. Through extensive numerical study we characterize these attractor loop shapes including their length, velocity, kink, and cusp distributions. We find that an initial loop containing $M$ harmonic modes will, on average, split into 3M stable loops. These stable loops are approximately described by the degenerate kinky loop, which is planar and rectangular, independently of the number of modes on the initial loop. This is confirmed by an analytic construction of a stable family of perturbed degenerate kinky loops. The average stable loop is also found to have a 40% chance of containing a cusp. We examine the properties of stable loops of different lengths and find only slight variation. Finally we develop a new analytic scheme to explicitly solve the string constraint equations.