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This work focuses on the problem of robotic picking in challenging multi-object scenarios. These scenarios include difficult-to-pick objects (e.g., too small, too flat objects) and challenging conditions (e.g., objects obstructed by other objects and/or the environment). To solve these challenges, we leverage four dexterous picking skills inspired by human manipulation techniques and propose methods based on deep neural networks that predict when and how to apply the skills based on the shape of the objects, their relative locations to each other, and the environmental factors. We utilize a compliant, under-actuated hand to reliably apply the identified skills in an open-loop manner. The capabilities of the proposed system are evaluated through a series of real-world experiments, comprising 45 trials with 150+ grasps, to assess its reliability and robustness, particularly in cluttered settings. The videos of all experiments are provided at https://dexterouspicking.wpi.edu/. This research helps bridge the gap between human and robotic grasping, showcasing promising results in various practical scenarios.