In this paper, we present a systematic Monte Carlo study of the self-assembly of nonionic, amphiphilic, chainlike molecules in dilute solution. The focus is on the regime in which the molecules form relatively weakly segregated micelles, which are in equilibrium with small submicellar aggregates. We study the size and shape distributions of the aggregates, and the structure of the aggregates' cores and surfaces. In some cases, spherical micelles, relatively large nonspherical micelles, and submicellar aggregates, all coexist. The size distributions of the spherical micelles are approximately Gaussian, while the nonspherical micelles contribute non-Gaussian tails at relatively large aggregation numbers. The simulation results are interpreted in terms of a simple theory of spherical micelles, and the size distributions are compared with its predictions. For the cases where the agreement is good, we combine the simulations and the theory to calculate the critical micelle concentration as functions of the chain lengths and solvent quality. In cases where there are nonspherical aggregates, the asphericity is quantified using the principal radii of gyration of the micelles, and the size distributions are compared with mean field predictions that account for both spherical and nonspherical aggregates.