No output from shift register
Your problem is your multiplex drive.

First, though, I have to point out that I believe your problem description is wrong. When you say, "the shift registers have no voltage across any of the output pins", I'm pretty sure that you mean that no LEDs are lighting, and that is not the same thing at all.

Second, your schematic is extremely misleading, since you show single LEDs in your displays. In fact, each display segment is composed of 4 pairs of LEDs connected in series, and this makes a big difference,

With that said, when you use NPN for your display selectors, you produce the following circuit for a selected segment:







simulate this circuit &ndash; Schematic created using CircuitLab

Since Q1 is being used as an emitter follower, the LED Voltage cannot be more than about 4.3 volts, and so the LEDs are not lighting.

You need to replace your multiplexing transistors with an NPN-PNP pair, like so:







simulate this circuit

Driving Q1 will cause base current to be pulled through Q2, turning it on. You do not need a 100 ohm in this case, 1k will be more than adequate, and the 2k base resistor will draw about 5 mA through the base, which will be more than adequate to turn on Q2 to the tune of 20 mA or less. You could actually use 5k and the circuit would work perfectly.