Missable!
There are 3 snails for each investigation. This will have to be done in one complete playthrough, as there is no chapter select or back tracking to previously visited areas. The locations for all the snails can be found below.
First investigation:
Snail 1: The first snail is in the Candy Kingdom. After you sneak into the Kingdom through the sewers, a fight will occur. Soon thereafter there is a cutscene with Princess Bubblegum. After the cutscene it will be immediately to your right.
Snail 2: When you first gain entry to Ron James' Shop, it will be to your right near the fireplace.
Snail 3: When you get to the lowest level of Ron James' basement (when you go through the manhole), it will be sitting on a pipe near the far right of the area.
Second investigation:
Snail 4: When you get to Lumpy Space Princess's camp in the forest it will be sitting on a broken washer sticking out of the ground.
Snail 5: When you exit the Candy Palace the snail will be sitting on the table closest to the palace stairs near the bottom of the screen.
Snail 6: Near the very end of this investigation the Snail will be sitting at the bottom of the stairs in the Fiona and Cake archives. DO NOT click on the Lumpy Space Princess before you try and click the snail as this will end the investigation.
Third investigation:
Snail 7: When you reach Magic Man's house derelict house, this snail is by the front tire of the green car.
Snail 8: Before you fight the third boss you will be helping the Lumpy Space Princess by turning a giant speaker. The snail will be sitting in plain sight near the bottom of the screen.
Snail 9: As soon as you break out of Lemongrabs Cell, he will be sitting on the top of a barrel nearby at the northern end of the room.
Fourth investigation:
Snail 10: This snail is in the Fire Kingdom core. After you raise the second bridge you will see him sitting on the wall below the scaffolding. When you approach it, you'll trigger a combat sequence. After you complete the combat you can interact with it.
Snail 11: This one is a bit harder to find. When you are in the cave, to get a quest item, you will have to use Jake to be lifted up to two platforms in one section. On the top section you will interact with a blue crystal to get a quest item. The snail is next to this blue crystal. It is in the section where you have to get a wand from a golden dragon's mouth.
Snail 12: At the very end of this investigation you will burn a bill of sale. BEFORE YOU DO THIS, go to the yard sale table farthest from Marceline. The snail will be sitting on top of the table.
Fifth investigation:
Snail 13: When you go back to the Wizard City, the Snail will be sitting on the stairs of the shop between Ron James' shop and the vending machine.
Snail 14: This snail is in the Palace of the Ice King. The snail is located in the bottom of the Ninja Armory Room, the same location where you used underwear to fire old pizza at the rebelling penguin.
Snail 15: This snail is sitting in plain sight immediately to the left of the Flame Princess when you first go to recruit her into your band.