Here's the guide for Devil May Cry HD that helps you out with Blue Orb pieces and Secret Missions, as both are tied in ways. Once again, read ahead at your own risk, as the below may contain some spoilers.
SECRET MISSIONS:
*** Spoiler - click to reveal ***
Secret Mission 1 - Critical Hit
Mission 3. To get to this mission, you have to first kill Phantom, then go back to the Broken Bridge area and deliberately jump down. That will activate the Secret Mission.
Objective: You have to kill the Sin Scissors with one blow. One way to do this is to use the Shotgun at range to stun it: The blast should hit it in the scissors and only knock it off balance, and not damage it. When it's stunned, run in to point-blank range and blast it again in the face. This should do it. Devil Trigger might help if you can't get it the first time around.
If you can't master this technique, another one to try is to wait for it to attack. Jump over the attack or jump back, such that you're stil face-to-face with it, and attack it with your sword from far enough away that you just hit the scissor, not the Sin's face. This should cause it to stagger and put you a step closer to it. Activate Devil Trigger and step forward while aiming your shotgun. Blast it as it's recovering. If your timing is right, you will fill its face full of buckshot just as it tuns its mask towards you.
Because the area you're in drains your health, you will have an issue with surviving multiple attempts at this secret mission. Using Devil Trigger will not only help you kill the thing quickly, but it will halt your health loss. As you wait for the Sin Scissors to drift down into your range, take this time to taunt it to fill your DT gauge. You can fill up to four runes before it even gets close enough to attack, provided you use double-handed taunts. This way you will always have full Devil Trigger for the fight.
NOTE: Completing this mission rewards you with Can You Keep A Secret?
Secret Mission 2 - Phantom Baby
Mission 4. At the start of Mission 4, turn around and go right back into the Cathedral where you fought Phantom. You now have 30 seconds to kill 100 Phantom Babies. Go!
Objective: There are several clusters of the little insects scattered throughout the room. Remember to equip Ebony & Ivory before you enter, since the bugs die in one hit. Even stepping on them is sufficient to kill them. Start stomping and blasting. Don't worry about killing every last one of them; there's more than 100 in the room. Just start near the door, kill the majority in one cluster, then move to the next, firing all the time. If you still can't do it, a Holy Water will clear the room in less than 1 second.
If you're strapped for Red Orbs, and you don't care about your Devil Hunter Rank at the end of the mission, you can farm this Secret Mission. Each Phantom Baby will drop 1 Red Orb, and you will get over 100 in the room with each attempt. There is no limit to the number of attempts you can make. Enter the room and kill slowly, making sure you run out of time before you reach 100 kills. Kill everything, collect the orbs, leave, then try again.
Secret Mission 3 - Phantom Baby 2
Mission 4, After clearing Secret Mission 02, go to the opposite end of the long corridor from Secret Mission 02 and enter the room with the Divinity Statue.
Objective: Without using weapons or items, you have to step on 100 Phantom Babies in 30 seconds. Just keep running up and down the narrow space and you should win. It does take a little luck, though, as the Phantom Babies will wander around and spread out a bit. If you fail, keep trying. Eventually you'll get enough spider guts on your soles to win.
As a tip, if you're having trouble with this one, using Devil Trigger doesn't break the rules of the mission, so the faster run speed provided by Alastor's DT can make this mission much simpler.
Secret Mission 4 - The 3 Beasts
Mission 04, after defeating the Shadow, go to the room with the airplane and ride the elevator down again to activate this Secret Mission. There is only one shot at this, so if you fail and want to complete this mission, don't save. It's unfortunate but you'll have to reload your previous save.
Objective: Preparation is key. Purchasing Air Raid, Stinger, an Untouchable, and a few Blue and Purple Orbs are all highly recommended.
All you have to do is fight three Shadows and survive. Focus on defense, and take pot shots at them with whatever firearm you like until one reveals its core, then do your best to melee that one while dodging the others. Using Devil Trigger will help prevent the other Shadows from interrupting your attacks and will ramp up your damage, but watch your health and always stay on the defensive.
One strategy to defeat them was to walk in with Ebony and Ivory, wait for their ranged shadow spike attacks, enable Devil Trigger and blast away as rapidly as possible. They're stuck while they're "spiked" and you do a nice chunk of damage. Using Devil Trigger while slashing away at them is also a good idea so determine your best approach. The above information is still absolutely critical.
Before starting this mission, take out Phantom and as he's dying, keep taunting his corpse to rebuild Devil Trigger.
Secret Mission 5 - One-Eyed Evil
Mission 7. Where the Old Key is found in Mission 6, in the pipe room. You can do this during Mission 7 or in the interval between Mission 7 or Mission 8, however; if you do the latter, it will not count towards your Devil Hunter Rank at the end of the mission. If done during Mission 7, your health will constantly decrease because of the Guiding Light you're holding.
Objective: Let one Kyclops kill the other. Despite what the secret mission description says, you only need one Kyklops to kill the other: After that, you can polish off the second on your own. You can stand between them, jump when they attack, and hope that one eventually kills the other with their rock-spit attack. Another, safer way to do this is to jump onto the pipes jutting out of one side of the room, and let them blunder around until one dies. Then jump off and kill the second one.
Secret Mission 6 - Wandering Ghosts
Mission 11. This Secret Mission is located in the last area of the previous mission. Go back through the doors where you first start Mission 11.
Objective: The Blue Orb Fragment is just hanging in mid-air, waiting for you to take it. The only way to get to it is to use the Enemy Step ability and jump off off the Sin Scissors's heads. After bouncing off of them, you then can use Wall Hike off of the wall to get even more height. Eventually you'll get it, but it takes one part skill and two parts luck.
NOTE: If you are going in this order, you should unlock The Secret Six after completing this Secret Mission.
Secret Mission 7 - Water Cell
Post-Mission 13. Return to the Captain’s Room on the ghost ship and check out the Captain’s skeleton.
Objective: Defeat all 8 enemies (Blades) underwater.
Secret Mission 8 - Treasure Of The Reaper
Mission 15. Check the picture wall in the rolling-spike trap room from Mission 14.
Objective: Defeat all enemies (3 Shadows).
Secret Mission 9 - Shadow Of Darkness
Mission 16, after defeating Nightmare. Airplane room.
Objective: Locate and defeat all the hidden Shadows. (2 in Airplane Room, 1 in Main Foyer upstairs across broken path, 1 in portculis Room)
Secret Mission 10 - Stairway Of Tranquillity
Mission 16 or Mission 17, after defeating Nightmare. In the Coliseum.
Objective: Several Sargasso will be conveniently arranged in a staircase type pattern and, most importantly, will not attack you on Normal or lower difficulty. You have to jump up their heads to the top. It may take a few tries, but you'll get there with patience.
If you find yourself overshooting the orb by using the last skull, air raid may be of assistance.
Secret Mission 11 - Blue Gem In The Sky
Mission 17, on the Broken Bridge. To activate the Mission is the tricky bit, though. Move from the dead end to the door end without falling once, over the floating pieces. Jumping onto the door end will automatically activate the Secret Mission.
Objective: You will be presented with a Death Scythe, a broken bridge, and a floating Blue Orb fragment. There are a few possible strategies to reach the orb fragment:
Avoid the Death's attacks and utilize the Jump Pads he scatters for you. This takes patience and luck, and some resetting, as he scatters the pads randomly and you will more than likely end up in the drink a few times.
Kill the Death, jump on the arch above the door to the cathedral, and use Air Raid to get to the blue orb fragment halfway across the bridge.
Once the secret mission starts assuming you used Air Raid to cross the broken bridge use a Devil Star fight Death utilize the Jump Pads and aim as high as possible activate Air Raid and head to the Blue Orb ignoring Death once you have the Blue Orb finish him off defeating him will charge your DT use Air Raid a third time to cross the bridge back.
Secret Mission 12 - Hidden Bangle
Mission 21. To find this mission, you have to examine the wall directly across from the Divinity Statue that's half-claimed by organic material in Mission 21. It is in the first room you start in.
Objective: Walk down the path, killing the Nobodies and collecting a Blue Orb along the way. At the end is a room with a T. Rex skeleton in it. This one doesn't spit fireballs like the one in Mission 17, only belches fire like a flamethrower, and must be defeated with good old-fashioned elbow grease. Any Devil Arm works well, but it is impervious to gunfire. After the "dragon" is dead, two Frosts will spawn (Shadows on Easy). Use Ifrit to kill them. There is an unconventional Red Orb Crystal between the dragon's legs as well. Break that to net some extra orbs.
Once everything is dead, moving platforms will spawn. You must jump across these to reach the Bangle of Time.
NOTE: If you are going in this order, you should unlock The Secret’s Out after completing this Secret Mission.
Strategy for Shadows:
The Shotgun or Ebony & Ivory are most effective for exposing the Shadow's core. Nightmare B is not recommended against Shadows, as it drains DT energy. Ifrit is also useful against Shadows; especially if you plan to deal a great deal of damage quickly, or you are in a confined space. The quickest way to finish off a Shadow is to simply activate Devil Trigger with the shotgun equipped. Once its core is revealed, use the Kick-13 combo while in Devil Trigger. If you prefer Alastor, you can use the Stinger combo. Waiting for its spear form is a good strategy, and may leave the core open to attack. However, on DMD mode this is simply a waste of time unless you're certain you can expose the core without waiting too long. Once enough damage has been done, Shadows will turn red and attack with even greater ferocity. If they catch you in their jaw-like form while in this phase, they will explode kamikaze style and deal a great amount of damage.
Facing off against more than one Shadow can be challenging, particularly if you have not fully upgraded Dante. However, it is vital to keep an eye on all of them if possible. If you only focus on one, you would leave yourself vulnerable to attacks from the others. Their corkscrew attack is both accurate and destructive, hence you need to know where they are at all times. Going into Devil Trigger can help here. By using Air Raid, try to force as many Shadows as you can into their core form, then use Vortex to hit all of them at once. Overall, using Devil Trigger can be a lot quicker to end a fight with these demons than ordinary attacks.
Avoiding the Corkscrew attack can take some practice, although you can simply roll to the side to dodge it or jump over it as the demon comes back down. However it can be a bit difficult to dodge, especially if you don`t see it coming (such as when fighting multiple Shadows). On harder difficulties this does massive damage so practicing dodging it is essential. The spike attack from the ground is not too difficult to dodge as long as you are prepared for it. If you're fighting more than one Shadow, it's best to be wary of the situation around you rather than the spikes.
On Dante Must Die mode, be quick about defeating them. If you are attempting "The Three Beasts" on DMD mode, make sure you stay far away from them while they're in DT mode. In this mode, the Shadows will automatically release their magical energy as opposed to firing it after being hit by Dante`s sword, with devastating results.