From here it looks like the 12v rails are pulsing over 100V. Have you set the probe attenuation setting correctly? Try a 10 ohm load on the 5V rail.
I noticed it was ticking on each spike (although there doesn't appear to be a relay anywhere in the PSU). I don't have a resistive load box in my setup, and a spare 1/4W resistor would swiftly result in a puff of magic smoke, so I attached a 13.5V incandescent lamp between the 5V rails. The ticking stopped and it lit the bulb nicely with a steady +5V output. However, the +12V was giving out +17.2V. I understand this is unloaded, but that's quite a difference. Does this sound normal?
Next step is to wire the synth up and see if it powers up. I have a 5V to 3.3V converter here, but I don't have anything to attach to the 10V rail; or the SW - although I'm assuming it won't need the SW pin attached as that is possibly just feeding the switch information back to the PSU. Maybe the +10V is only needed for the switch?
I guess if it doesn't power up without the +10V attached, I could always inject +10V from my bench supply. If that works I'll throw in a 7810 to feed it...
I'll be back with the results in due course...