We present experimental observations at comparatively low supercooling of morphology transitions from dendritic to faceted structures in polymer crystals growing in thin films of a poly-2-vinylpyridine-block-polyethyleneoxid copolymer. Our results are compared with theoretical concepts describing morphological instabilities of single crystals. Although these concepts originally were not developed for polymers, they allow to describe and interpret our experimental results quite well. In particular, the measured temperature dependence of the width W and frequency of dendritic side branches and the radius of curvature p of the growth tips of the crystals follow these concepts. We present preliminary evidence for the influence of polymer attachment kinetics and reorganisation processes behind the growth front. Polymer thin films provide valuable model systems for studying general concepts of crystallisation and allow to distinguish at which point the connectivity of the crystallising units within chain-like molecules starts to play a measurable role.