Open Radio Access Networks (O-RAN) enhance RAN flexibility, interoperability, and intelligence by introducing open interfaces, disaggregation of RAN components, and datadriven control. This flexibility has the potential to significantly reduce capital and operating expenses (CAPEX/OPEX) by enhancing efficiencies in network energy consumption. Advanced Sleep Modes (ASMs) are an advantageous energy-saving technique for 5G base stations (BS), with a set of sleep depths that progressively deactivate different RAN components. The deployment and effectiveness of ASMs in O-RAN raises a number of challenges that have yet to be investigated. This paper surveys and critically reviews RAN ASMs, as well as other energy-enhancing techniques applied in O-RAN. It presents the first granular, percomponent O-RAN Radio Unit (O-RU) power model, that will enable researchers to quantify the power savings gained from deploying ASMs. A calculation of the energy savings achieved at each ASM level per O-RU component for different 3GPP functional splits is provided. The power consumption of O-RU sub-components for functional splits 6, 7.2x, and 8 is shown. The default 3GPP functional split 7.2x as adopted by O-RAN shows that substantial power savings can be achieved by applying ASMs to the O-RU, achieving a minimum of 80% savings depending on the choice of SM as compared to power consumption at full load. Finally, a discussion of the open research challenges and opportunities in deploying ASMs in O-RAN is provided.