Ch.1, After (or during-whichever you prefer) the foggy fiend mission on Mushroom Rock Road, return and follow the path down to the Den of the Woe, you will automatically pick up Crimson Sphere 9, which shows as Crimson Report 1 on Shinra's list.

Ch.2 Again, return to the Den of Woe, you will be given Crimson Sphere 7
Ch.2 Crimson Sphere 10 is in Logo's room, (Part of the story, I believe.)

Ch.3 Go to Guadosalam speak with a few people, when finished go to the "circle " bedroom. You'll find it hidden up top. Crimson Sphere 4, (I believe that this is the only one that you more that one shot to get.)

Ch.3, When Djose is a hotspot, GO TO BEVELLE, follow who you see underground, (you'll need to lower the lift again, and go to where you fought the aeon) then after scenes, you'll need to pick up the sphere on the floor. Crimson Sphere 1

At the End of Ch. 3 Crimson Sphere 2 & 3 are given to you.

Ch. 4 after the concert go to the engine room and LeBlanc will give you Crimson Sphere 5. You will be told she is there if you didn�t skip the scenes, (Do not speak to Shinra until you have it.)

In Ch. 5
Crimson Sphere 6 is in Bevelle on Floor 1. Crimson Sphere 8 is on Floor 20.