Crystallization processes are in general sensitive to detailed conditions, but our present understanding of underlying mechanisms is insufficient. A crystallizable chain within a diblock copolymer assembly is expected to couple curvature to crystallization and thereby impact rigidity as well as preferred morphology, but the effects on dispersed phases have remained unclear. The hydrophobic polymer polycaprolactone (PCL) is semi-crystalline in bulk (T(m) = 60°C) and is shown here to generate flexible worm micelles or rigid vesicles in water from several dozen polyethyleneoxide-based diblocks (PEO-PCL). Despite the fact that `worms' have a mean curvature between that of vesicles and spherical micelles, `worms' are seen only within a narrow, process-dependent wedge of morphological phase space that is deep within the vesicle phase. Fluorescence imaging shows worms are predominantly in one of two states - either entirely flexible with dynamic thermal undulations or fully rigid; only a few worms appear rigid at room temperature (T << T(m)), indicating suppression of crystallization by both curvature and PCL hydration. Worm rigidification, which depends on molecular weight, is also prevented by copolymerization of caprolactone with just 10% racemic lactide that otherwise has little impact on bulk crystallinity. In contrast to worms, vesicles of PEO-PCL are always rigid and typically leaky. Defects between crystallite domains induce dislocation-roughening with focal leakiness although select PEO-PCL - which classical surfactant arguments would predict make worms - yield vesicles that retain encapsulant and appear smooth, suggesting a gel or glassy state. Hydration in dispersion thus tends to selectively soften high curvature microphases.