[SOLVED] Using SPI on Arduino Nano to control a shift register (TPIC6B595)

Other Hardware LEDs and Multiplexing
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No, SPI does not operate the SS line in master mode. That's because you're free to connect multiple slaves (and thus need multiple SS lines) to a single master.

I big difference I see, with shiftOut() you make latch LOW before you start sifting and with SPI only after. I don't know if the TPIC6B595 cares but I do know that

  digitalWrite(latchPin, LOW);
  digitalWrite(latchPin, HIGH);

Makes a very very very small pulse. Probably to small.