Sleep is a prominent physiological state observed across the animal kingdom. Yet, for some animals, our ability to identify sleep can be masked by behaviors otherwise associated with being awake, such as for some sharks that must swim continuously to push oxygenated seawater over their gills to breathe. We know that sleep in buccal pumping sharks with clear rest/activity cycles, such as draughtsboard sharks (Cephaloscyllium isabellum, Bonnaterre, 1788), manifests as a behavioral shutdown, postural relaxation, reduced responsiveness, and a lowered metabolic rate. However, these features of sleep do not lend themselves well to animals that swim nonstop. In addition to video and accelerometry recordings, we tried to explore the electrophysiological correlates of sleep in draughtsboard sharks using electroencephalography (EEG), electromyography, and electrooculography, while monitoring brain temperature. The seven channels of EEG activity had a surprising level of (apparent) instability when animals were swimming, but also when sleeping. The amount of stable EEG signals was too low for replication within- and across individuals. Eye movements were not measurable, owing to instability of the reference electrode. Based on an established behavioral characterization of sleep in draughtsboard sharks, we offer the original finding that muscle tone was strongest during active wakefulness, lower in quietly awake sharks, and lowest in sleeping sharks. We also offer several critical suggestions on how to improve techniques for characterizing sleep electrophysiology in future studies on elasmobranchs, particularly for those that swim continuously. Ultimately, these approaches will provide important insights into the evolutionary confluence of behaviors typically associated with wakefulness and sleep.