Here's all the locations of the upgrades:
Light speed shoes-In the sewer by the burger shop.
Crystal ring-In the hotel door above the manager behind the counter (Hold B in front of the door).
Ancient light-On a rock to the left of Red Mountain.
Jet anklet-In the sewer behind Twinkle Park.
Rhythm badge-When you enter the past turn left and go up the stairs and turn right.
Shovel claw-Beside Kiki the monkey in a cave by the hillside of Tails workshop.
Fighting gloves-On a ledge above big the cat's house.
Warrier feather-Beat Robotnik's high score on whack-a-sonic.
Long hammer-Beat your new score twice.
Big The Cat:
1)In the ice cap level.
2)In a cell onboard the egg carrier.
3)In a sewer behind Twinkle Park.
4)In a cave in the Aztec Forest.
Life Belt-Turn to the left when you walk through the altar to Ice Cap.
Power Rod-Lift up Big's bed and jump in.
Jet booster-In the ammunition room on the egg carrier.
Battle gun-On the opposite side of the ammunition room.