First remove socketed and potentially corrosion damaged parts and use a continuity tester, to make sure traces weren't corroded into open circuit. Look at the schematic, and for each CPU pin find somewhere else on the board it's connected to and make sure there's continuity.
Eyeball for bad caps - leaky or skanky looking.
Then power it on and check all supply voltages with a volt meter. Probe various locations around the board.
Next, time for the scope: probe the oscillator to make sure it produces a clock. With passive probes you will likely only see a signal on the driven side of a crystal. That's alright.
Then check the reset - make sure it comes out of reset okay. Scope the reset line while powering on, or hitting a reset button.
Then probe EPROM control signals: CE#, OE# to make sure attempts are made to read from it.
If that's okay and you still see no signs of life, make sure there is a video signal.
If that's present, and it's just blank, then remove the EPROMs and read them to make sure they're okay. If not, get some compatible EEPROMs and replace the originals. A cheap and simple EPROM reader/programmer that will do the job is the TL866.
If it's still not working, use a continuity tester to verify all bus signals are correct. If there are buffers scope both sides of the buffer along with a OE# control (you need three channels for this) to make the output tracks the input when it should. A logic analyzer can be used for this, too.
If it executes code, all looks fine, then it could be a peripheral is dead, for example it might not prompt you for a floppy if it doesn't detect a floppy drive. (Don't know anything about Amigas, sorry.)