During the 2nd act, you will a mission from Yuri about a mystery box on a roof. This mission unlocks Taskamaster’s bomb and drone challenges. After completing a couple of those, the stealth and combat challenges unlock. Spectaculars are not required for this trophy. If you’re not going for spectacular scores, you can just do the main objective for each challenge type.
Combat: Just clear all enemies any way you chose. These challenges are the easiest and you should get spectaculars without any real effort.
Stealth: For these you just need to take down all enemies without ever being spotted. Being spotted by an enemy is an automatic failure.
Bomb: Just clear all bombs within the time limit. After locating bombs, hit L1 & R1 to throw bombs. Sometimes they’re hidden under objects. Hit L1 & R1 to clear objects. If you’re going for Spectaculars, you need to spam R1 after throw each bomb to cover it in webbing and complete as fast as possible for greater time bonus.
Drone: Just follow the drone to it’s destination before time expires. You can ignore the blue spheres if you’re not going for spectaculars. If you’re going for spectaculars, you want to grab most or all blue spheres for points.