We introduce a motion forecasting (behavior prediction) method that meets the latency requirements for autonomous driving in dense urban environments without sacrificing accuracy. A whole-scene sparse input representation allows StopNet to scale to predicting trajectories for hundreds of road agents with reliable latency. In addition to predicting trajectories, our scene encoder lends itself to predicting whole-scene probabilistic occupancy grids, a complementary output representation suitable for busy urban environments. Occupancy grids allow the AV to reason collectively about the behavior of groups of agents without processing their individual trajectories. We demonstrate the effectiveness of our sparse input representation and our model in terms of computation and accuracy over three datasets. We further show that co-training consistent trajectory and occupancy predictions improves upon state-of-the-art performance under standard metrics.