You need to post both your schematic and code, use the correct code tags. If you don't know what that means read the how to use this forum sticky post.
Have you not got a schematic? It is difficult to follow a physical wiring diagram. The bit round the transistor does not make any sense. What should it be?
Kreplach:
When I connect ULN2003 GND to logic ground it does not light at all.
Then you have something else wrong.
You can not switch a 12V anode like that. You need to source current through the anode with a PNP transistor.
That is still not a schematic so don't say you posted one when you have not.
When I connected the shift register ground to the 12 volt ground path instead of the 5 volt ground back to the arduino it works as expected.
I will work on furnishing better schematics so that the community will be able to offer help without having to guess at what it is I am trying to accomplish.