Delineation of curvilinear structures in images is an important basic step of several image processing applications, such as segmentation of roads or rivers in aerial images, vessels or staining membranes in medical images, and cracks in pavements and roads, among others. Existing methods suffer from insufficient robustness to noise. In this paper, we propose a novel operator for the detection of curvilinear structures in images, which we demonstrate to be robust to various types of noise and effective in several applications. We call it RUSTICO, which stands for RobUST Inhibition-augmented Curvilinear Operator. It is inspired by the push-pull inhibition in visual cortex and takes as input the responses of two trainable B-COSFIRE filters of opposite polarity. The output of RUSTICO consists of a magnitude map and an orientation map. We carried out experiments on a data set of synthetic stimuli with noise drawn from different distributions, as well as on several benchmark data sets of retinal fundus images, crack pavements, and aerial images and a new data set of rose bushes used for automatic gardening. We evaluated the performance of RUSTICO by a metric that considers the structural properties of line networks (connectivity, area, and length) and demonstrated that RUSTICO outperforms many existing methods with high statistical significance. RUSTICO exhibits high robustness to noise and texture.