Spontaneous formation of locally ordered patterns during dewetting of thin films on homogeneous and heterogeneous substrates is investigated based on the 3-D nonlinear equation of motion. Physicochemical heterogeneities engender the rapid formation of the primary holes that serve as "seeds" for the formation of locally ordered structures. The secondary multiring structure surrounding the primary hole evolves by one of the following two different pathways depending on the film thickness vis-A-vis the location of the minimum in the spinodal curve: (A) Thick films evolve by the formation of secondary satellite holes that originate from a ring-like depression behind the rim of the primary hole. The process of ordering is repeated until the true spinodal holes appear on the remaining substrate. (B) Ordering in a relatively thin film occurs by the formation of droplets caused by the disintegration of the elevated rim that surrounds the primary hole. The radial distance between the successive ordered layers is close to the spinodal length scale, lambda(m). Droplets within the same layer are separated by a distance slightly greater than lambda(m), whereas holes within the same layer are separated by a distance slightly less than lambda(m). The number density of holes or droplets in the ordered pattern is of the same order as the predictions of the spinodal theory. The number of ordered layers and the size of the locally ordered domain depend significantly on the relative magnitudes of the time scales for the following events: (1) formation of the primary hole, (2) growth of holes (inverse of hole-growth velocity), (3) formation of a secondary feature (hole or droplet) adjacent to the primary hole, (4) true spinodal rupture far from the primary hole. The morphology of an ordered structure can therefore be tailored by modulation of the film thickness and the short- and long-range intermolecular interactions (substrate surface properties), since these affect the time scales 1 to 4 in different ways.