My own simple Secret Missions guide for Dante:
Secret Mission 1 (Mission 3)
Requirements: Devil Trigger
Just beat all the enemies within the time limit. Use Beowulf and Rebellion to kill enemies faster. If you are stuck here, come back later when you have Devil Trigger so that you can kill them much more faster.
Secret Mission 2 (Mission 5)
Requirements: Gunslinger (any level), Ebony and Ivory (recommended Level 3) or any other favourite firearms
Just shoot the four Enigmas using your firearms from faraway. Constantly dodge Enigma's projectile attacks. If you are stuck here, come back when you have fully upgraded your firearms and try again.
Secret Mission 3 (Mission 7)
Requirements: Air Hike (Rebellion, Beowulf, Agni and Rudra), basic mastery on Enemy Jump, Air Raid (Nevan), full Devil Trigger bar, complete Purple Orbs
You need to stay up on the air for 20 seconds. First, lure enemies to your side, then jump. Try to do Enemy Jumps on enemies, and Air Hike to give yourself more time. Another option is to use Air Hike, then use Air Raid to stay on the air for more than 10 seconds. You can even use Devil Stars to replenish your DT while you are using Air Raid.
Secret Mission 4 (Mission 8)
Requirements: Gunslinger style (Level 3), Shotgun (Level 3)
You need to ride up an elevator to obtain the Blue Orb Fragment. Once you ride the elevator and when the enemies appear, use Fireworks to clear off all enemies quickly. As enemies are in DT form, you need to do this quickly. If there are too enemies on the elevator, the elevator will go down and you need to try again.
Secret Mission 6 (Mission 10)
Requirements: Full Devil Trigger, complete Purple Orbs, Air Raid (Nevan)
You need to collect all 40 Red Orbs within the time limit. Just use Air Raid and collect all the Red Orbs on the air. Remember to collect all of the White Orbs on the way to replenish your DT.
Secret Mission 7 (Mission 11)
Requirements: Full Devil Trigger, complete Purple Orbs, Stinger Level 2, Devil Stars (Optional)
You need to collect the Blue Orb Fragment within the time limit. Just use DT and run as fast as possible to the Blue Orb Fragment. Once your DT runs out and you are near to the Blue Orb Fragment, use Stinger to quickly close in to the enemy nearby to the Blue Orb Fragment.
Secret Mission 8 (Mission 13)
Requirements: Agni and Rudra, Beowulf
You need to kill all of the enemies. Use Agni and Rudra to kill all the Arachnes easiily, and use Beowulf to kill the Hell Vanguards.
Secret Mission 9 (Mission 13)
Requirements: Gunslinger (level 3), Ebony and Ivory (level 3)
You cannot let any enemies alive as you progress. Use Ebony and Ivory, and tap the Square button very fast to destroy enemies quickly.
Secret Mission 10 (Mission 16)
Requirements: Basic mastery of puzzle-solving skills.
You just need to ensure the light emitted from one statue hits another statue. Break any mirrors that block the light path whenever necessary. There are several variations of this puzzle whenever you retry the Secret Mission, so observe carefully.
Secret Mission 11 (Mission 17)
Requirements: Trickster (recommended level 3)
You need to get through the spikes to get the Blue Orb Fragment. Use Sky Star for easy passage on that room, with proper timed jumps as well. Move accordingly to the area where the spikes do not touch as you move forward.
Secret Mission 12 (Mission 18)
Requirements: Trickster (level 3)m Air Hike
You need to travel upwards to get the Blue Orb Fragment. Observe carefully where the moving platforms, and use any necessary moves to travel upward. You can always use either Air Hike or Sky Star whenever you use Wall Hike.
You can have a look at the complete guide here:
http://www.gamefaqs.com/ps2/920607-devil-may-cry-3-dantes-awakening/faqs/36512
Haven't done all of the Secret Missions as Vergil, so I can't guide you about it.
