Constructing a robust and discriminative local descriptor for 3D shapes is a key component of many computer vision applications. Although existing learning-based approaches can achieve good performance in specific benchmarks, they usually fail to learn sufficient information from shapes with different types and structures (e.g., spatial resolution, connectivity, and transformation). To solve this issue, we present a more discriminative local descriptor for deformable 3D shapes with incompatible structures. Based on spectral embedding using the Laplace--Beltrami framework on the surface, we construct a novel local spectral feature that exhibits high resilience to changes in mesh resolution, triangulation, and transformation. The multiscale local spectral features around each vertex are then encoded into a geometry image called vertex spectral image in a compact manner. Such vertex spectral images can be efficiently trained to learn local descriptors using a triplet neural network. Then, we present a new benchmark dataset for training and evaluation by extending the widely used FAUST dataset. We utilize a remeshing approach to generate modified shapes with different structures. Furthermore, we evaluate the proposed approach thoroughly and conduct an in-depth comparison to demonstrate that our approach outperforms recent state-of-the-art methods on this benchmark.