Where to find the seven chaos emeralds in shadow the hedgehog

One of the things that makes "Shadow the Hedgehog" different from other Sonic games is that you don't actively collect the Chaos Emeralds. The first Emerald (green) is collected during the first level: Westopolis. It's mandatory that it is collected; you can't get any further without it. The other Emeralds automatically end up with Shadow at the end of the game, no mattter which ending you get. They aren't collected as part of the gameplay, so no Special Stages for you.