The latch which drives the transistor could be bad. It's listed as A6/7 on the schematics and the part number is CD4099B.

Here's the data sheet for the latch:
http://www.mouser.com/ds/2/405/cd4099b-452623.pdf

It's also possible that one of the signals driving the latch is not getting there for whatever reason.

I would expect that during game play pins 5, 6, and 7 (IOBA0,IOBA1,IOBA2) should be strobing all the time. Pin 2 (Reset) should be high, and pins 3, and 4 (IOBD0, and IOSOLWRI) will be strobing when the CPU is updating the latch to turn the flippers on or off. I would start by probing these signals with a logic probe and then possibly a logic analyzer, but you could also start by shotgunning the latch and possibly fix it.

Atari_Superman_Flipper_Relay_Driver_Schematic.jpg

That's awesome news. Glad to hear it!

I was hoping that would be the solution because I suspect it would have gotten more tricky if you had to work back from there.