I have this same thing working with my Arduino Uno a few months ago and tested a version with a nano on a breadboard.
I am trying to shift data into a 595 shift register and display it via LEDs using Shiftout. I have digitally probed my circuit board, but there is nothing coming out of the outputs. I have tested the inputs (Latch, Clock, and Data) and they seem to be getting signal.
Here is one major difference: my old project showed a dimly lit LED when I connected to the latch, but on my new circuit board with nano, it is just bright. I have used delays in the code to see it visibly blinking.
Am I making any sense? Probably not. I am very tired and frustrated. I have verified continuity between all the pins. The picture of the PCB is attached.
New development: If the chip is very loose in the socket, it will kinda, almost work.