Summary: The enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores \(U: \mathbb {R} ^ 2\times S ^ 1 \to \mathbb {C}\) were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores \(U: \mathbb {R} ^ 3 \times S ^2 \to \mathbb C\). First, we construct the orientation score from a given dataset, which is achieved by an invertible coherent state type of transform. For this transformation we introduce 3D versions of the 2D cake wavelets, which are complex wavelets that can simultaneously detect oriented structures and oriented edges. Here we introduce two types of cake wavelets: the first uses a discrete Fourier transform, and the second is designed in the 3D generalized Zernike basis, allowing us to calculate analytical expressions for the spatial filters. Second, we propose a nonlinear diffusion flow on the 3D roto-translation group: crossing-preserving coherence-enhancing diffusion via orientation scores (CEDOS). Finally, we show two applications of the orientation score transformation. In the first application we apply our CEDOS algorithm to real medical image data. In the second one we develop a new tubularity measure using 3D orientation scores and apply the tubularity measure to both artificial and real medical data.