The Light Speed Shoes are an upgrade to Sonic the Hedgehog in Sonic Adventure 2. They are a pair of shoes that allow Sonic to perform the Light Speed Dash and travel along trails of rings. They are the equivalent to Shadow's Air Shoes.
The upgrade is found in the Metal Harbor stage. After the first Star Post, get over to the next platform. The Light Speed Shoes can be found on the top of the tower on that platform, which can be reached with the pully.
The Light Speed Shoes are only required to complete Crazy Gadget. Because of this they are non-optional but they can be skipped up until Crazy Gadget. If Sonic were to jump onto the boxes surrounding the pulley after it has pulled him up, he can jump up onto the roof of the structure were the pulley is located. Going to the edge closest to the Shoes, charging up a Spin Dash and letting go will send Sonic off at high speeds, and at the other edge with perfect timing jumping will send Sonic up in the air fast enough to reach the barge that would normally be inaccessible. There is another line of rings at the end of this barge to be used to travel to the next section, but jumping down from the barge reveils a platform with a rocket allowing Sonic to get across. After this the level has no more uses for the Light Speed Dash other than attacking Shadow on Final Rush.