Any suggestions on how to identify the power problem?
Start by measuring the outputs of the power supply to see if any are at the correct voltage. If some are good and some are not that narrows down where to look. If none are, either something early in the power chain is not working, or there are interlocks that turn off all rails if any one is out of regulation.
Look for any blown fuses, look for electrolytic capacitors which have a bulging top or have any traces of electrolyte leakage. Bad electrolytic caps are a common source of problems on cheap equipment. Look for any broken solder joints, especially around heavier components like large capacitors, transformers, etc. that might have resulted from a drop to a hard surface or high vibration (being carried around in a truck for example).
I do not know the internals of an ADA8000, look around the power supply to see if there are any stickers or labels indicating it may be from a third party. If it is an off-the-shelf supply you might be able to find information online.
Sometimes you can disconnect the power supply from the rest of the circuitry to check whether a short downstream is preventing the power supply from turning on, but that doesn't always work with switching supplies, some need a minimum load to run and won't power on correctly with the load disconnected.
I just found a picture online of the internals of an older model. Does yours look like the one in this picture?
ADA8000 teardown picture
If so you are in a good starting position, that is a plain linear power supply, no tricky switching power supplies to debug. Start following connections through components, and draw yourself a schematic diagram to keep track of everything, then start measuring voltages through the circuits. It is a little easier if you have access to an o'scope, but you can do a lot with just a multimeter. With AC power removed check the diodes, make sure none are shorted or open. Check the caps for bulges as mentioned previously. Apply power and measure across the caps, make sure whether power is getting through the diode bridge and charging up the caps. Measure AC voltage across the caps, that should be a very low value (e.g. on a 5V DC supply I would expect less than 0.5V AC and around 8V DC, or less than 1V AC on 15V for the 12V DC supply).
If the supplies are OK at the caps, check the outputs of the linear regulators, possibly a regulator was underspecified or has an undersize heatsink. Touch the heatsinks carefully to see if a regulator is heating up really quickly, that would indicate either a blown regulator, or a short circuit downstream from the regulator.
That should give you a good start, there are only so many things that can go wrong with a linear power supply. I don't know what voltages are present, probably +/-12V or +/-15V for the analog components, 3.3V for the digital components, and either an extra winding from the transformer, or a diode tripler type circuit to generate 48V for the microphone phantom supply. Possibly a +5V analog for the A/D converters, you would just have to look up some component specs to see what to expect. If you are lucky there will be labels on the PCB for the various voltages, otherwise you will just have to start measuring.