Despite the ubiquity of missing data in real-world datasets, there is a dearth of research on how missing data impacts feature selection stability. To address this gap, we present the Feature Selection Stability Evaluation framework for High-dimensional data (FSEH), which encompasses four key components: data missing process, data imputation, feature selection, and evaluation. The framework is tested across four typical high-dimensional classification datasets, employing five classic feature selection methods, and utilizing five metrics for a holistic comparison. Our study reveals that all tested methods experience an evident decline in stability as the proportion of missing data increases, and different methods exhibit different sensitivity to data missing. Notably, we discover unexpected performance improvements in specific scenarios, linked to mean imputation compatibility. This work establishes a dual evaluation framework combining stability and classification metrics, providing a critical guide for balancing performance and robustness in feature selection under data missing.