Weeeell if you have a scope then a whole world of options opens up to you.
Of course you should always start by checking that the amp is actually getting 12V to start with, its the only component on the board using the 12V feed so the lack of sound would be the only impact from losing the 12V. Then move on to an eyeball check of the amp and sound subsystem area of the board, look for pins bent on the underside of the board, these can short out proximal components. Look for missing components, especially capacitors as these often stand fairly proud of the board and are often bashed about. Look for scratches that may have cut the tracks and any signs of damage. If chips are socketted then pulling them out and reseating can fix bad connections due to oxidisation of pins. While the chip is out its wise to look for rust in the chip sockets, double check for bent pins on the chip and make sure you dont cause any when putting the chip back.
After the sanity check you can start from either end and work inwards.
1) From the Amp
Find out what the Amplifier chip is and google the datasheet, this will give you the pinout so you can then find which is the audio input pin. Poke your scope at that and you should see the audio signal if its there - bear in mind that the dipswitch settings on the board may well set it to be quiet in attract mode so you may have to coin it up and start a game to see anything. You can also see if there is anything on the output pin which is helpful if you have a cabling issue, not unheard of with games needing jamma harnesses.
If you have a signal but no sound it will be one of the following
a) Bad amp chip
b) Dried out electroyltics in the amp section.
c) Smashed or missing capacitors in the amp section.
d) Faulty volume control pot.
If you have no input signal then you need to work back through the circuit to the DACs, usually small 8 pin chips, find out what DACs are on board and pull up their datasheets. You will need to check their inputs with the scope to see if they are getting any work to do, if they are and the outputs are silent then you need to see if the chip is actually enabled, the datasheet should help you there. If you get to that point and the DACs have no input I usually find its worth cutting your losses and starting at the other end.
2) From the CPU
Actually I always hit the CPU clock pin before I do any of the above but anyway...
Start at the clock pin, which on a Z80 is pin 6, you should see a regular string of pulses on this pin, a CPU with no clock will do absolutely nothing. If you have no clock then thats likely to be the fault, if you have a clock then you move on.
Second to hit is the Reset and Halt pins (26 and 18 respectively), these are active low pins and usually drawn as RESET or HALT with a line drawn above them to denote "active low". As reset will be active when the pin is low you will need the pin to be high for the CPU to even run, if its low then you need to find out why, if its pulsing then the board is trying to kick start the CPU as it had detected the CPU is doing nothing, this is the watchdog circuit, some boards have it and some don't, Golden Axe doesnt from memory. Halt is an input and output pin, it needs to be Low or the CPU will just stop. Its usually used to stop the CPU in the event of a bus error, tho the CPU can halt the board as a result of software I think.
Next with the scope is to hit up the address lines, A0 to A15...
...if the CPU is running you should see activity on these lines, any lines that are doing nothing can indicate a fault, either with the CPU or the bus it is trying to drive. Bear in mind that not all address lines may be used by the program running, so if you have an upper few lines doing nothing that may not be a problem, ie if A0 to A12 are active and A13-A15 silent that can be normal. However there should be no holes, so A0-A7 ok, A8-A10 silent and A11-A15 active is almost certainly a fault. If you see this and the CPU is socketted then its just simplest to pull the chip and test it elsewhere, or drop in another CPU and try again.
If you are feeling brave you can test you have connectivity between the CPU, the RAMs and the ROMs on all address lines although there are some caviates about how things are wired up that may this difficult to describe, ie TTL logic may well form part of the addressing system so its highly board specific.
If thats OK then you go on to the data lines, these are bidirectional pins, data outbound to the RAM and data inbound from the RAM/ROM. Its work looking at the state of the signals if they are a blocky mess (if they look like a city skyline) it can indicate bad RAM, also you should have all the datalines on the CPU active, you are highly unlikely to see any missing lines on a RAM that is connected directly to the CPU.
That should get you started!
Never let it be said that I give short answers
