Part segmentation is a crucial task for 3D curvilinear structures like neuron dendrites and blood vessels, enabling the analysis of dendritic spines and aneurysms with scientific and clinical significance. However, their diversely winded morphology poses a generalization challenge to existing deep learning methods, which leads to labor-intensive manual correction. In this work, we propose FreSeg, a framework of part segmentation tasks for 3D curvilinear structures. With Frenet-Frame-based point cloud transformation, it enables the models to learn more generalizable features and have significant performance improvements on tasks involving elongated and curvy geometries. We evaluate FreSeg on 2 datasets: 1) DenSpineEM, an in-house dataset for dendritic spine segmentation, and 2) IntrA, a public 3D dataset for intracranial aneurysm segmentation. Further, we will release the DenSpineEM dataset, which includes roughly 6,000 spines from 69 dendrites from 3 public electron microscopy (EM) datasets, to foster the development of effective dendritic spine instance extraction methods and, consequently, large-scale connectivity analysis to better understand mammalian brains.