The obvious test for shorted turns is the resistance measurement. Another method is to view the winding as an inductor and form a crude resonant circuit and apply a current pulse. You then count the number of 'rings' - you could use a PC & sound card as a oscilloscope. If there are shorted turns then the circuit will not 'ring' as much if at all. This technique I have seen used on the line output transformers of TVs and monitors as these have a large turns ratio so the resistance test is normally not too conclusive. I dare say the currents and voltages involved in this method should make testing a lot safer. Sounds like a job for a micro - apply pulse and count the rings, repeat a few times, average the result and display.

Food for thought.