The first thing to try is of course a new CPU, since it's socketed and easy to swap for a known good one. If you have a spare 060 laying about :-)
Careful not to damage the socket when removing the device. Ideally you should use a PGA extraction tool.
A majority of the Cyberstorm (mk3) I've repaired have had bad solder joints between the socket pins and the PCB.
One way you can sometimes test for this is to apply downward pressure on top of the CPU while you power it on. If the computer boots, there's a 95% chance it's a socket soldering problem.
About the only reliable way to fix this is to replace the socket.
Removing it is easy enough using a hot air SMD rework tool and a large nozzle.
Fitting a new SMD socket is not a simple task without special equipment, so you can just make your own using those individual pin sockets which are soldered into PCBs for leaded devices.
Each one is hand soldered to the SMD pads. They must be perfectly aligned, or you'll have trouble seating the 060 in place.
It's pretty tricky to get perfect, so you need to be well experienced with both SMD and hand soldering.
Considering the mk2 uses the same socket, this could be the problem with yours? The mk2 seemed more reliable overall somehow. Phase 5's soldering process would appear to be a lot better on those boards.
Also check obvious things, like status of the MPU's reset line (should be high), you have an output from the crystal oscillator, you have known good SIMMs fitted (I think the card needs at least 4Mb or something??).
Failing that, send to Jean-Jaques at Amiga.Fr for repair.
http://www.bourgogne-informatique.com