Linear recurrent neural networks enable powerful long-range sequence modeling with constant memory usage and time-per-token during inference. These architectures hold promise for streaming applications at the edge, but deployment in resource-constrained environments requires hardware-aware optimizations to minimize latency and energy consumption. Unstructured sparsity offers a compelling solution, enabling substantial reductions in compute and memory requirements--when accelerated by compatible hardware platforms. In this paper, we conduct a scaling study to investigate the Pareto front of performance and efficiency across inference compute budgets. We find that highly sparse linear RNNs consistently achieve better efficiency-performance trade-offs than dense baselines, with 2x less compute and 36% less memory at iso-accuracy. Our models achieve state-of-the-art results on a real-time streaming task for audio denoising. By quantizing our sparse models to fixed-point arithmetic and deploying them on the Intel Loihi 2 neuromorphic chip for real-time processing, we translate model compression into tangible gains of 42x lower latency and 149x lower energy consumption compared to a dense model on an edge GPU. Our findings showcase the transformative potential of unstructured sparsity, paving the way for highly efficient recurrent neural networks in real-world, resource-constrained environments.