Neutral models (NMs) are built to test null hypotheses and to detect properties at work in an object or a system. While several studies in geomorphology have used NMs without explicitly mentioning them or describing how they were built, it must be recognized that neutral models more often concerned theoretical explorations that drove such use. In this paper, we propose a panel of NMs of river (channel) networks based on a well-established relationship between observed and simulated sinuosity properties. We first simulated new instances of river networks with a (one-parameter) neutral model based on optimal channel networks (OCN) and leading to homogeneous sinuosity watersheds. We then proposed a "less neutral" model able to generate a variety of river networks accounting for the spatial heterogeneity of observed properties such as elevation. These models, providing confidence levels, allowed us to certify that some properties played a role in the generation of the observed network. Finally, we demonstrated and illustrated both models on the Bidasoa watershed (Spain-France frontier), with a new dedicated software (called SSM). NMs in geomorphology ensure to progressively help to identify the process operating in an observed object, and to ultimately improve our understanding of it (i.e. intrinsic need). But they also provide simulated samples statistically "similar" to an observed one, thus offering new alternatives to every process carried by the observed object (i.e. extrinsic need). Artificial river networks studied here would be of great value to environmental sciences studying geomorphology and freshwater-related processes.