Quoted from Quench:

It's wax that's leaked out from the transformer running warm/hot.
Measure the output voltages to determine if it's still any good and in spec. Running LEDs instead of incandescents will reduce the strain on the transformer if you're concerned.

Quoted from Tuukka:

Many years ago, I made a simple transformer/coil tester to quickly check a coil. It is based on the coil (or transformer) inductance. If a coil has DC voltage across it, and thus current flowing, disconnecting voltage makes the coil try to keep current flowing and a reverse voltage appears across the coil.
This simple circuit shows the voltage across coil with a red LED lighting when the button is pressed. When the button is released, the red LED turns off, but there will be a short flash of green LED, as the coils stored magnetic field turns into reverse voltage, indicating the coil is OK.
This will not work with all possible coils, but for most pinball coils and transformers, it is OK.
[quoted image][quoted image]

Quoted from Tuukka:

The human eye is quite good in picking up the short flashes of green led. Anyway the energy stored in a coil can be quite large, and give a longer flash.
The AC transformer works by magnetically (via the iron core) connecting primary and secondary windings. Voltage (or current) change in primary winding changes its magnetic field, which in turn generates voltage in the secondary winding. Of course, with AC power, the voltage changes continually at 50/60 cycles per second, making the secondary winding to follow it. This causes a voltage to appear across secondary winding, depending on the ratio of primary and secondary coils.
If the primary has 1000 turns, and secondary 100 turns, then the voltage across secondary winding is 1/10 of the primary. If the primary is connected to 115 volt AC, there will be 11.5 volts at the secondary.
However, in my tester, all coils, whether in transformer or pinball solenoid can be handled just as a single coil. Basically, draw current through a coil, then stop it and the coil shows reverse voltage.

Quoted from Tuukka:

This is just a simple go/no go tester.
For more accurate testing of transformers, a good way is (with an unconnected transformer) to feed AC voltage into one known good secondary and then check the voltages on all the other secondaries and primary winding.
Many pinball transformers have a 6.3V G.I. secondary. Connecting it to 6.3VAC source, you should be able to check all the other voltages, even the primaries.

Quoted from Tuukka:

The transformer doesn't care which is its primary or secondary.