ewen:
_two power supplies – one 15V (thick red line) and one 5V (thin red line) sharing a common ground:
_running the arduino off the 5V supply
_running the shift registers off a clean 5V supply from the arduino (thin dashed red line)
change that 5v on Vin to something over 7 volts.
read the data sheet on Vin.
http://arduino.cc/en/Main/arduinoBoardUno
someone already said to add a resistor on the LED when you connect it properly.