You have probably ruled out a problem with the USB connection, so the remaining possibilities are faulty USB cable and a problem on the shuffle.
There are separate connection points for data and power, so it is possible for the shuffle to charge, while not being able to connect to sync data. If the shuffle's light did any blinking at all, there is a data connection. But since it never blinks, the shuffle seems to only see the connection for power, not data.
To rule out a faulty USB cable, you'll need to try your shuffle with another USB cable or use the cable you have with another shuffle. If you can't do that with what you have, and there is an Apple Store nearby, you may want to make a reservation at the store's Genius Bar (the store's tech support). Take your shuffle there, along with your USB cable.
If you can rule out the USB connection and the USB cable as causes, the problem is likely to be on the shuffle. It may be related to the headphones jack, or the logic board. NOTE: If it has been less than one year since the purchase date, it is under warranty.