Coreset selection methods have shown promise in reducing the training data size while maintaining model performance for data-efficient machine learning. However, as many datasets suffer from biases that cause models to learn spurious correlations instead of causal features, it is important to understand whether and how dataset reduction methods may perpetuate, amplify, or mitigate these biases. In this work, we conduct the first comprehensive analysis of the implications of data selection on the spurious bias levels of the selected coresets and the robustness of downstream models trained on them. We use an extensive experimental setting spanning ten different spurious correlations benchmarks, five score metrics to characterize sample importance/ difficulty, and five data selection policies across a broad range of coreset sizes. Thereby, we unravel a series of nontrivial nuances in interactions between sample difficulty and bias alignment, as well as dataset bias and resultant model robustness. For example, we find that selecting coresets using embedding-based sample characterization scores runs a comparatively lower risk of inadvertently exacerbating bias than selecting using characterizations based on learning dynamics. Most importantly, our analysis reveals that although some coreset selection methods could achieve lower bias levels by prioritizing difficult samples, they do not reliably guarantee downstream robustness.