Here is the full list of all 47 Raiden IV OverKill trophies - there are 38 bronze, 4 silver, 4 gold and 1 platinum.
Obtained all trophies.
Clear first round with double play!（OVERKILL MODE/Difficulty NORMAL or higher）
Clear LEVEL 3 with double play!（OVERKILL MODE/Difficulty NORMAL or higher）
Clear ULTIMATE difficulty with 1 credit.（OVERKILL MODE/No DUAL）
Clear VERY HARD difficulty with 1 credit.（OVERKILL MODE/No DUAL）
Clear HARD difficulty with 1credit.（OVERKILL MODE/No DUAL）
Clear NORMAL difficulty with 1 credit.（OVERKILL MODE/No DUAL）
Clear LEVEL 1 with double play!（OVERKILL MODE/Difficulty NORMAL or higher）
Earn a score of over 20,000,000!（All game modes/Any difficulty/No DUAL）
Resucue all Miklas in LEVEL 2-1 - LEVEL 2-7.（OVERKILL MODE/SCORE ATTACK MODE/Any difficulty/No DUAL）
Destroy all Sols in LEVEL 1 - LEVEL 7.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty NORMAL or higher/No DUAL）
Clear BOSS RUSH MODE!（No continue/Difficulty NORMAL or higher/No DUAL）
In Double play, Clear first stage using only the 2P ship without shooting from 1P aircraft.（All game modes/Difficulty NORMAL or higher）
Make boss self destruct by running out of time.（All game modes/Any difficulty/No DUAL）
Clear ARCADE MODE .（No continue/Difficulty NORMAL or higher/No DUAL）
Clear with only Proton Laser.（OVERKILL MODE/Continue is acceptable/Difficulty EASY or higher/No DUAL）
Clear with only Plasma Laser.（OVERKILL MODE/Continue is acceptable/Difficulty EASY or higher/No DUAL）
Clear with only Laser.（OVERKILL MODE/Continue is acceptable/Difficulty EASY or higher/No DUAL）
Clear with only Vulcan.（OVERKILL MODE/Continue is acceptable/Difficulty EASY or higher/No DUAL）
Destroy more than 1,000 meteorites in mission stage 2!（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Rescue all fairies in mission stage 1!（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Clear with 100% of destruction rate in LEVEL 7.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Obtain the 1UP item in LEVEL 4.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Collect every ship in the Gallery!
Destroy the LEVEL 2-7 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 2-6 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 2-5 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 2-4 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 2-3 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the Jeeps in LEVEL 3.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Destroy all warships that appear in LEVEL 6.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Clear with every Score Ring collected from the enemies in LEVEL 2.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Destroy the LEVEL 2-2 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 2-1 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-7 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-6 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-5 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Accumulate 7 bombs and use all of them at once.（All game modes/Any difficulty/No DUAL）
Destroy all tanks that appear in LEVEL 5.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）
Destroy more than 12 enemies at once with one bomb.（All game modes/Any difficulty/No DUAL）
Clear any single LEVEL with one ship and no bombs.（All game modes/Any difficulty/No DUAL）
Destroy the LEVEL 1-4 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-3 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-2 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Destroy the LEVEL 1-1 Boss.（OVERKILL MODE/Difficulty NORMAL or higher/No DUAL）
Rescue the Fairy in LEVEL 1.（OVERKILL MODE/SCORE ATTACK MODE/Difficulty EASY or higher/No DUAL）