As an emerging biometric technology, finger vein recognition has attracted much attention in recent years. However, singlesample recognition is a practical and longstanding challenge in this field, referring to only one finger vein image per class in the training set. In single-sample finger vein recognition, the illumination variations under low contrast and the lack of information of intra-class variations severely affect the recognition performance. Despite of its high robustness against noise and illumination variations, sparse representation has rarely been explored for single-sample finger vein recognition. Therefore, in this paper, we focus on developing a new approach called Progressive Sparse Representation Classification (PSRC) to address the challenging issue of single-sample finger vein recognition. Firstly, as residual may become too large under the scenario of single-sample finger vein recognition, we propose a progressive strategy for representation refinement of SRC. Secondly, to adaptively optimize progressions, a progressive index called Max Energy Residual Index (MERI) is defined as the guidance. Furthermore, we extend PSRC to bimodal biometrics and propose a Competitive PSRC (C-PSRC) fusion approach. The C-PSRC creates more discriminative fused sample and fusion dictionary by comparing residual errors of different modalities. By comparing with several state-of-the-art methods on three finger vein benchmarks, the superiority of the proposed PSRC and C-PSRC is clearly demonstrated.