I started writing then it suddenly occured to me that I didn't know what style monitor we are talking about LCD or CRT ..
The first thing that comes to mind as dai says is capacitors especially in LCD/TFT monitors. The problems start in the power supply region and might be caused by faulty electrolytic capacitors in the power supply or Inverter part of the power supply.
Most LCD that I have seen when opened up either have very obvious faulty capacitors or comprise of two boards, one being power supply & inverter board whilst the other being the VGA interface. Since you have so many units it will very easy to swap boards between a good and bad unit to see which board is faulty. In my opinion the power supply / inverter boards are easily repaired since most components are easily found .. start by replacing all electrytic capacitors. 99.9% of the time this is the problem .. having said that though there are some LCD monitors that have separate inverter board and a small surface mount fuse which goes open circuit. If you can recognise what components are what, look for it and see if it is intact. Should it be open circuit you'll have to figure out what value it is and replace it. I tend to replace with a standard style fuse on flying leads that I have insulated totally with heat-shrink tubing so that it cannot under any circumstances short out anywhere or cause damage.
In CRT monitors however , loss of picture usually boils down to loss of high voltage at the anode of the tube
IT'S ~25000 VOLTS don't try measuring it !!!! at least without the proper environment and test equipment .. a DVM will usually only go to ~1000 Volts !!
or loss of heater voltage.
Some monitors use a separate heater supply from the primary power supply transformer whilst others might have the heater voltage as part of the High Voltage Transformer circuit.
You'd have to follow back the heater supply line to see which type of circuit the monitor is using.
If you know anything about electronics and diagnosing faults it will probably not be too difficult for you to sort this out
IF YOU HAVE NO ELECTRICAL / ELECTRONIC KNOWLEDGE KEEP YOUR FINGERS OUT AND SEEK PROFESSIONAL ADVICE
THE VOLTAGES INSIDE ELECTRICAL / ELECTRONIC EQUIPMENT ARE LETHAL, USUALLY KILLING EITHER THE INQUISITIVE INEXPERIENCED INTRUDER, THE INQUISITIVE BYSTANDER WHO WANTS TO POKE HIS FINGER INSIDE OR BOTH WHEN THEY GET IN EACH OTHERS WAY!!!!
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