Modern processors are increasingly adopting tensor cores as key computational units. Compared to existing designs for dense and structured sparsity, recent dual-side sparse tensor cores have evolved to support general sparsity. However, existing methods still face limitations on generality (incomplete sparse kernel support prevents broad applicability) and performance (outer-product/row-row schemes yield unsatisfactory hardware utilisation, data reuse, and energy efficiency). In this paper, we propose Uni-STC, a unified sparse tensor core that delivers high-performance dataflows for four key sparse kernels: sparse matrix-vector multiplication (SpMV), sparse matrixsparse vector multiplication (SpMSpV), sparse matrix-multiple vector multiplication (SpMM), and sparse general matrix-matrix multiplication (SpGEMM). To efficiently support these diverse sparse workloads, we first introduce BBC, a unified sparse format co-designed with Uni-STC's dataflow. We then design UniSTC's architecture supporting (1) fine-grained task partitioning to improve resource utilisation, (2) parallel sparse-tile processing to enhance data reuse, and (3) a dynamic network to reduce intermediate data movement and energy consumption. Evaluated across 2893 SuiteSparse and 302 DLMC matrices, Uni-STC demonstrates significant improvements, outperforming the state-of-the-art RM-STC with a$2.21 \times$geomean speedup and$2.96 \times$higher energy efficiency.