Ample evidence supports a role for sleep in the offline consolidation of memory. However, circumstances exist where forgetting can be as critical as remembering, both in daily life and clinically. Using a directed forgetting paradigm, here, we investigate the impact of explicit cue instruction during learning, prior to sleep, on subsequent remembering and forgetting of memory, after sleep. We demonstrate that sleep, relative to time awake, can selectively ignore the facilitation of items previously cued to be forgotten, yet preferentially enhance recall for items cued to be remembered; indicative of specificity based on prior waking instruction. Moreover, the success of this differential remember/forget effect is strongly correlated with fast sleep spindles over the left superior parietal cortex. Furthermore, electroencephalography source analysis of these spindles revealed a repeating loop of current density between selective memory-related regions of the superior parietal, medial temporal, and right prefrontal cortices. These findings move beyond the classical notion of sleep universally strengthening information. Instead, they suggest a model in which sleep may be more ecologically attuned to instructions present during learning while awake, supporting both remembering and targeted forgetting of human memories.