Fibers that exhibit good elasticity or extensibility and tenacity, and low modulus are prepared from propylene-based copolymers. The propylene-based copolymers comprise at least about 50 weight percent (wt %) of units derived from propylene and at least about 8 wt % of units derived from one or more comonomers other than propylene, e.g., ethylene. Particularly preferred propylene copolymers are characterized as having 13C NMR peaks corresponding to a regio-error at about 14.6 and about 15.7 ppm, the peaks of about equal intensity. In one aspect of the invention, fibers are subjected to stress-induced crystallization by subjecting the fiber to tensile elongation during draw.