(click the recursive spoilers for even more hints)
Crystal 1: Located at the beach (starting area).
It's very close to the entrance to the sea caves.
Crystal 2: Located below the green/blue area east of the beach.
It's in an area behind a hidden passage, detector may help.
Crystal 3: Located at the brown/green ruins.
It's at a very high place, triangle jump required.
Crystal 4: Located at the silhouetted (black/yellow) ruins.
Entrance to this area is somewhat hidden (detector helps). Requires the eye.
Crystal 5: Located at the black&white area (west of the green forest).
Another area with a hidden entrance, detector may help. Umbrella is required as well.
Crystal 6: Located at red/lava area.
It's in a screen with two separate sides.
Crystal 7: Located at the blue silhouetted caves.
It's somewhere at the bottom, in a hidden tunnel.
Crystal 8: Located at the pink cloud area.
It's at the very top of the area. Requires the eye.
Crystal 9: Located at the blue/pink/brown forest.
It's behind a wall with a hidden passage, requires a rather tricky triangle jump.