Phenomena displaying the behaviour expected when molecular species segregated at the growth front lower the supercooling for continued growth have been observed using linear low-density polyethylene (LLDPE) nucleated on a fibre to form a macroscopic row structure. Initially, a close-packed array of lamellae grows epitaxially from the fibre, with the lamellae too close together for other parallel lamellae to form between them, so that rejected species can only accumulate ahead of the growth envelope. The more-branched molecular sequences rejected in this polymer lower the equilibrium melting temperature and with it the isothermal supercooling. In consequence, the texture coarsens and lamellae become thicker, both for the rows and in adjacent spherulites. During this process the isothermal growth rate declines continuously as the impurity layer develops, but spherulites always grow faster than rows because their more open crystal-melt interfaces give lower concentrations of impurities. The habit of the rows is then prone to change abruptly as spherulitic growth nucleates sporadically at points around the instantaneous growth envelope and grows faster than, and hence ahead of, the remainder of the row. The depth to instability and the average distance between spherulitic nuclei both increase with crystallization temperature but the former only slightly. It is suggested that nucleation occurs where fluctuations have reduced the concentration of segregated species and we found that the internuclear distance scales approximately as the inverse one-third power of the growth rate. No such instability or cellulation is found for spherulites growing adjacent to the rows even though their characteristic microstructure develops from the start, thereby confirming in a particularly clear way that morphological instability is not the cause of spherulitic growth in polymers.