Although the SG-1000 is a fairly hardy console, there are several conditions which could cause the system not to power on. Fortunately the system has fairly simple power circuitry and faults are easy to trace. Listed are the following components which could prevent the console from powering on along with possible remedies.
All versions of the SG-1000 have a single-sided motherboard, meaning that all of the traces and solder joints are located on only one side of the board (namely the bottom). With enough stress, the solder joints on the system's various connectors could become broken and work themselves loose. Check the power jack for any broken solder joints; if there are, reflow the solder joints and add a small amount of solder.
It is also possible for the inner contacts of the power jack to become corroded or oxidized, especially if the system has been stored in poor conditions. The SG-1000 uses a standard 5.5 x 2.1 mm barrel jack which is fairly easy to source.
The SG-1000's power switch is rather intricate and is an open design with several copper strips and small plastic pieces. This type of switch was also used in the Mark III and can be prone to failure, usually caused by the plastic pieces breaking off. This cannot be repaired so the only remedy is to completely replace the switch with a more modern type, demonstrated in this repair video on a Mark III.
7805 Voltage Regulator
Like so many other classic game consoles, the SG-1000 uses a 7805 voltage regulator to step down the incoming voltage from the AC adapter to the 5V that it requires for operation. However, the SG-1000 has no voltage input protection other than a 10 uF electrolytic filter capacitor to protect the console against the use of an improper power supply, which means that the 7805 will be one of the first components to fail. Furthermore, it is not too uncommon for 7805 regulators to fail with age and considering that the oldest examples of the SG-1000 are nearly 40 years old (as of this writing), the 7805 should be considered the most likely cause for a no power condition.
Thankfully, the 7805 is an extremely common and inexpensive part and is also fairly easy to replace with basic soldering and desoldering knowledge. Care must be taken though, as the SG-1000 motherboard is fragile and lifted pads or traces can result if one is not careful. The 7805 can also be replaced with a drop-in switching regulator, but this is not recommended as they can generate high amounts of EMI (electromagnetic interference) and can create interference in the video signal.