On my MacBook Air, I had to plug the iPod in on the left side to get it to be seen.
For some unknown reason I get a flickering light on the right side. So try a different USB port.
Maybe because I am on battery, and the right side gets less power to save batter?
Based on what I have read online about it, I am not alone in this behavior.
This could also be why it is not charging as well.
Have you tried a restart? Turn off the power, wait 10 seconds then turn it on?
Have you tried a different cable, maybe the one you have is bad?
Which version is it out of curiosity? I have a 2nd generation.
My iPod charges using the headphone jack.My cable is that old too, and works.
Your iPod Shuffle may be dead, not charging, not getting power.
If you have a friend with iTunes, I would try it on their system to be sure it is not your system. If they have a cable too, that would be even better.
So does mine. You should look at finding another system to test with.
The fact that your light never stops blinking can indicate that your charging system (batter or circuitry to the battery) is broken.