The abundance of sparse matrix formats, vendor libraries, and heterogeneous hardware architectures creates a complex optimization landscape in which performance depends on the interplay among data representation, software implementation, and hardware capabilities. To address this complexity, we present Situla, a comprehensive characterization of sparse-dense and sparse-sparse matrix multiplication across Intel Xeon CPUs, NVIDIA H100 GPUs, and many-core ARM Neoverse processors. Our systematic evaluation of CSR, CSC, BSR, and COO formats demonstrates that hardware superiority is not absolute but is instead contingent on software efficiency and matrix structure. Notably, a custom OpenMP implementation on ARM CPUs can outperform vendor-optimized libraries and achieve GPU-level performance for a significant subset of workloads. By analyzing latency, throughput, and power efficiency across diverse matrices from the SuiteSparse collection, Situla challenges conventional assumptions regarding accelerator dominance and highlights critical trade-offs between format selection, platform choice, and energy consumption. Situla’s findings can serve as a blueprint for developing sparsity-aware runtime systems capable of intelligent, dynamic workload scheduling in largescale heterogeneous environments.