I don't see any decoupling on the 595 chips.
They need 100n ceramic caps from VCC to ground, close to each chip.
Made something similar a while ago, and I wrote a HIGH to the three shift register pins BEFORE setting them to OUTPUT with pinMode.
Leo..
digitalWrite(latchPin, HIGH);
digitalWrite(dataPin, HIGH);
digitalWrite(clockPin, HIGH);
pinMode(latchPin, OUTPUT);
pinMode(dataPin, OUTPUT);
pinMode(clockPin, OUTPUT);
What is the value of the base resistor.