Electronic – How to connect multidigit 7seg common anode/cathode display to cascading shift register

My guess would be this:

To switch the display on:

• STB goes low
• Clock in the byte 00001101 (0x0D)
• STB goes high

To write the segments:

• STB goes low
• clock in the Write command: 00000000 (8 bits of zeros). This means: it's a write command, with auto-incrementing addresses, writing to address 0x00 on page 0x00.
• Now you can clock in the data to fill the memory. This consists of six bytes, one for each digit.
• STB goes high

To clock data (make sure the clock is slower than 1MHz):

• STB goes low
• Loop for 8 bits of the byte
  • Set data out bit
  • Clock goes low
  • Clock goes high
• End of Loop
• STB goes high

Here is a possible mechanism for the fault. You say it works (dimly) without the transistors, but doesn't work with them. So it sounds like the fault is in the transistors. Is there some reason that the transistors might be letting some current through when they shouldn't?

Yes. You're using a PNP transistor. As you know, these transistors are on when the base voltage is lower than the emitter voltage. They are off when the base voltage is higher than or equal to the emitter voltage.

The problem with the shift register chip is that the outputs are always lower than the emitter voltage. I couldn't quite make out the part number of the chip you're using, but according to the datasheet for the 74HC595 (page 6), the outputs don't quite reach Vcc. If there is a tiny voltage difference, then you could find that a small amount of current is leaking out of the base of the PNP transistor. With a gain of about 100, you could find that there's enough CE current to give noticeable light output on the LEDs.

Something to try: Add a schottky diode between Vcc and the emitter. This should drop the collector voltage by a fraction of a volt, just enough to allow the shift register to fully turn off the transistor.