Radar sensors employed for environment perception, e.g. in autonomous vehicles, output a lot of unwanted clutter. These points, for which no corresponding real objects exist, are a major source of errors in following processing steps like object detection or tracking. We therefore present two novel neural network setups for identifying clutter. The input data, network architectures and training configuration are adjusted specifically for this task. Special attention is paid to the downsampling of point clouds composed of multiple sensor scans. In an extensive evaluation, the new setups display substantially better performance than existing approaches. Because there is no suitable public data set in which clutter is annotated, we design a method to automatically generate the respective labels. By applying it to existing data with object annotations and releasing its code, we effectively create the first freely available radar clutter data set representing real-world driving scenarios. Code and instructions are accessible at www.github.com/kopp-j/clutter-ds.