Forthcoming imaging surveys will increase the number of known galaxy-scale strong lenses by several orders of magnitude. For this to happen, images of billions of galaxies will have to be inspected to identify potential candidates. In this context, deep-learning techniques are particularly suitable for finding patterns in large data sets, and convolutional neural networks (CNNs) in particular can efficiently process large volumes of images. We assess and compare the performance of three network architectures in the classification of strong-lensing systems on the basis of their morphological characteristics. In particular, we implemented a classical CNN architecture, an inception network, and a residual network. We trained and tested our networks on different subsamples of a data set of 40 000 mock images whose characteristics were similar to those expected in the wide survey planned with the ESA mission Euclid, gradually including larger fractions of faint lenses. We also evaluated the importance of adding information about the color difference between the lens and source galaxies by repeating the same training on single- and multiband images. Our models find samples of clear lenses with ≳90% precision and completeness. Nevertheless, when lenses with fainter arcs are included in the training set, the performance of the three models deteriorates with accuracy values of ~0.87 to ~0.75, depending on the model. Specifically, the classical CNN and the inception network perform similarly in most of our tests, while the residual network generally produces worse results. Our analysis focuses on the application of CNNs to high-resolution space-like images, such as those that the Euclid telescope will deliver. Moreover, we investigated the optimal training strategy for this specific survey to fully exploit the scientific potential of the upcoming observations. We suggest that training the networks separately on lenses with different morphology might be needed to identify the faint arcs. We also tested the relevance of the color information for the detection of these systems, and we find that it does not yield a significant improvement. The accuracy ranges from ~0.89 to ~0.78 for the different models. The reason might be that the resolution of the Euclid telescope in the infrared bands is lower than that of the images in the visual band.