Here are some scope pictures from a transformer under various conditions.
The pictures show the primary current of the transformer.

Picture 1 is normal unloaded primary current with 120 VAC applied. You can
see the onset of core saturation.

Picture 2 is with 145 VAC applied to the primary. The core is well into
saturation.

Picture 3 is primary current with 120 VAC applied, but with a single
shorted turn of 21 gauge wire around the center leg.

Picture 4 is primary current with 120 VAC applied and with a single turn of
14 gauge wire around the center leg.

This test could help detect a shorted turn.