I don't have a schematic ready but I can show you some pictures:
Okay, it appears you have your logic backwards. You have all your LED anodes wired to +V through a current limiting resistor. Then you are driving the individual cathodes from the output of your shift register. So, if you want to turn on an LED you must write a LOW into the shift register output so that current will flow from +V toward a low voltage(effectively ground). When you write a HIGH into the shift register output, you effectively have +V at both ends of your circuit, and no current flows. Does that make sense?
It absolutely does. Thank you very much. Problem solved.