睢 is not used independently in modern Chinese. Its old verb sense 'to stare / gaze up' survives only in the literary phrase 暴戾恣睢 (cruel and willful). Today it is met mostly in place names like 睢县 and the Sui River.
register
Classical / proper-noun only — seen in old idioms and geography, not in everyday speech.
Right side 隹 (short-tailed bird) is the sound element, drifting from zhuī to suī. It carries no 'bird' meaning here. The same phonetic also drives 虽 and 雎.