In the area where the fight with Nelo Angelo happens. Jump to the area above the entrance (where the fight ends) and go left to the end with the wall. The tower on the right has a blue orb fragment. Jumping to the left tower (just off-camera), then back, will allow you to reach (requires Air Raid or Air Hike). Alternatively, kick-jumping off the wall and holding jump while pushing towards the direction will allow you to grab the ledge and climb up (tricky to get the right position, but no abilities needed).
Beneath the headstone. In the area where you first encounter Blades, find the pathway to the right of the entryway, (when looking at it. It will be Dante's left as he enters the space.) Climb the pathway and strike the headstone a few times to expose the orb fragment beneath it.
On the pedestal in the greenhouse. In the room where you fight the Death Scythe, double jump over the glass wall. Walk over to the hole and fall down, then jump to the ground floor and the orb fragment is on the pedestal.
In the hold beneath the captain's room. After exiting the captain's room through the ventillation hole, swim down the stairs, and turn completely around. Head down the hole. It is floating at the far end, guarded by Blades.
Cliff in the waterfall area. Use Stinger from the top of the elevator while facing away from the waterfall. Air Hike may also be used to simply double jump onto the cliff.
On a ledge across the bridge in the tiered cliff. After defeating the Blades, enter the tunnel halfway down the cliff and climb to the top. Cross the rope bridge and jump down the right-hand side onto the ledge with the fragment.
The second fragment in the waterfall area, reached via Air Hike.
Secret Mission 8. Examine the wall of pictures in the spiked area. Will require you to backtrack into Mission 14 territory.
Behind a breakable wall behind the door with the blue shield. Go down the stairs after obtaining the Luminite and kill the Fetishes. The breakable wall is on the left. Smash everything in the room to find the blue orb fragment.
In the pit with the Blades (the area under the lances in the RED shield area) before obtaining Nightmare Beta or Pair of Lances. After defeating the Blades, kick off the wall under the item and the Blue Orb will appear on the pedestal. Alternatively, before striking the two seals and activating the pillars, you can use Air Raid to fly over to the pedestal. You will have to time air raid just right to get under the cross-beams.
At the center of the highest point of the Coliseum. Before the fight with Griffon, use the magical teleporters at the sides of the Coliseum to go to the above floor, then climb the stairs to get to the top. Now run across the path to the center. The fragment will appear and drop when you reach the center blue circle on the glass floor.
Inside a barrel in the last prison cell on Dante's left.
On a platform in the water column room. Climb the stairs until rubble blocks your path. The jut out is near there. The camera makes spotting this one difficult (the orb won't appear until you stand on the ledge); if you're having trouble finding it, you can also simply fill the shaft with water and swim up slowly until you spot the ledge.
In a secret hallway off of the room where you start. DO NOT step on the portal left by Nightmare yet. There is a second story to Dante's left. Kick jump up there; you may have to run towards the camera until the camera angle changes in order to see the break in the path to get there. From there, run towards the back of the room and zone through the door. Follow the trail of Red Orbs to the fragment at the end of the hall, can also be done in mission 18 before fighting nightmare.
Master Bedroom. It is beside the bed, on the opposite side of the Divinity Statue (you can get this in mission 17 instead - when you go back to the bridge for Secret Mission 11, jump through the right-hand portal).
Mirror World balcony area. This one requires the Air Raid skill(not really its accomplishable without) with Alastor and at least 3 DT Runes filled. Jump to the third story and you can see it just off of the turret across the way. DT and jump, and activate Air Raid when you're halfway in the air. If you activate Air Raid when you're at the height of your jump, you'll be too high, you may have to fly over it, turn Air Raid off, and hope that you pick it up on your way down. You can also jump onto the tower right behind the blue orb then drop down onto it. This may require Air Hike(or just use the wall next to this tower to jump on it).
Inside the tower after getting the previous fragment. Still requires Air Hike or Air Raid to get(nope neither is necessary since one can use again the wall next to it and jump up and hope you grab the ledge to get to the tower).