You can use this handy little tester to check for shorted out coils on an iron core xfmr.
Simply close the momentary contact switch, release and see if the neon light flashes. If it does, then you don't have any shorted coils. If it doesn't flash, then you likely have a shorted coil that's loading down the inductive flyback (or not enough inductance if it's a very small unit). As for open coils, simply use any continuity tester.
Another thing is simply physical inspection. Remove the covers (if any) then look and sniff. If you see discolored laminations, or evidence of melted insulation, or detect a burnt odor, then your unit has likely been overheated. Even if the coils didn't short, the unit should be discarded, as its core characteristics have likely been altered by overheating, and it probably won't last much longer anyway.
If all seems well, then test with the mains, and a light bulb in series. Once again, look, listen and sniff. Loud humming, crackling, wisps of smoke, smells like something burning, brightly lit bulb: something's not right.
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