Automated identification of the primary components of a neuron and extraction of its subcellular features are essential steps in many quantitative studies of neuronal networks. The focus of this paper is the development of an algorithm for the automated detection of the location and morphology of somas in confocal images of neuronal network cultures. This problem is motivated by applications in high-content screenings (HCS), where the extraction of multiple morphological features of neurons on large data sets is required. Existing algorithms are not very efficient when applied to the analysis of confocal image stacks of neuronal cultures. In addition to the usual difficulties associated with the processing of fluorescent images, these types of stacks contain a small number of images so that only a small number of pixels are available along the z-direction and it is challenging to apply conventional 3D filters. The algorithm we present in this paper applies a number of innovative ideas from the theory of directional multiscale representations and involves the following steps: (i) image segmentation based on support vector machines with specially designed multiscale filters; (ii) soma extraction and separation of contiguous somas, using a combination of level set method and directional multiscale filters. We also present an approach to extract the soma's surface morphology using the 3D shearlet transform. Extensive numerical experiments show that our algorithms are computationally efficient and highly accurate in segmenting the somas and separating contiguous ones. The algorithms presented in this paper will facilitate the development of a high-throughput quantitative platform for the study of neuronal networks for HCS applications.