Figment

There are 16 Memories in total to collect, but only 13 are required for this trophy (the individual memories associated with Time for Tea, Turning on the Light, and Waterboy are not required for this). You can check your progress on them by going to the very center of Cerebrum City. There is a large tree with a blue bench next to it, surrounded by 16 holes in the ground. Each time you find a memory, the pink memory orb will be in the hole it corresponds to. The order they're in is the following:
 

You will not be able to get them all in order unless you wait until the very end of the game to get them. 

The recommended order you collect these all in is as follows:
I, IV, V, VI, VIII, VII, II, III, X, XI, XIII, XII, XV, XIV, IX, XVI

Although only 13 are required for the trophy, all 16 locations will be listed below so if you are referencing just this trophy, you won’t need to do any backtracking for the three that have their own trophy(ies). Their locations are as follows:


Memory I: Outer Cerebrum 

Piper will make an audible cue that there is a memory, shortly after you use the Blue Battery to power the pencil bridge. Hit the button next to the pencil bridge to activate the moving platform just to below you. Go back up the bridge and grab the Blue Battery. Keep going up and hit the button until the middle bridge is active. Go down, past the house, and follow the platforms all the way until you are able to cross the moving platform. Use the Blue Battery on the empty cell to create another two bridges, allowing you to get the Memory. 

Memory II: Freedom Isles - Freedom Promenade

You can only get this after you have defeated the Nightmare in Freedom Isles. Once you have, you will automatically traverse back to Cerebrum City. Immediately turn around and head back to Freedom Isles. From the start, go up to the third Island and knock on the door to receive the memory.

Memory III: Freedom Isles - Freedom Promenade 
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You can get this before defeating the Nightmare in Freedom Isles, but it’s highly recommended you wait until after. This Memory will unlock Time for Tea and is not required for this particular trophy. Throughout Freedom Isles, you’ll need to knock on the door of 3 Kettle Houses (all of which are required for Seeking Inspiration) to turn their kettles on. One is located in Freedom Promenade (this will be the last one you turn on and is the 3rd image), one is located in Bright Planes (1st image), and the third is in the Gardens of Creativity (2nd image). Once you have the 2nd and 3rd on, it’s recommended you defeat the Nightmare and then return to Freedom Isles. After getting Memory I, continue down the path and take the Red Battery from the first red windmill you come across. Put it in the empty cell next to the house and knock on the door to receive the memory.

Memory IV: Freedom Isles - Freedom Promenade

After clearing the smoke from the Caterpillar, take the path up and at the top will be a large house with two doors. There will also be a handful of blacked-out windows. To get this memory, you need to get all the lights to be on at the same time (they will turn on one at a time). To get them to do so, you need to alternate knocking on the two doors. Start with the left door.

• Left Door
  • Response is “Breh!”, knock again
  • Response is “Bruh”, leave and go knock on the right door
• Right Door
  • Response is “Bruh”, knock again
  • Response is “Breh!”, leave and go knock on the left door

Repeat this process until all 9 lights have been turned on and the Memory will appear.

Memory V: Freedom Isles - The Bright Planes
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After you fix the Flute bridge, you’ll pick up a wheel that will enable you to turn the catapult. Turn it (so you fly to the right). When you land, cross either of the small bridges and go up the lift to a house with a Blue roof. Remember this house (it's the 2nd picture shown - you won’t be able to get the memory right away). When you complete Bright Planes, you’ll end up in the Overpass of Originality. You'll be able to see the Blue Battery you need right away (shown in the 1st picture) At the start, go left of the trumpet and keep heading left across the moving platforms. Use the acid from the Yellow Plague Rat to kill the flowers so you can restore power to the green windmill and continue down, across more platforms. When you reach the end, pick up the Blue Battery (Dusty will make an audible cue that you may have missed something). Take the lift down and head back to Bright Planes. Go all the way back to the house with the Blue roof. Put the Blue Battery in the empty cell and knock on the door to get the Memory.

Memory VI: Freedom Isles - Gardens of Creativity 

You only need 2 Seeds to actually complete this section, but you’ll need all 3 for the Memory (one is North of the green windmill, one is up a lift to the left of the Tentacle, and one is on the first little island as you progress towards the poison). You will also need the Green Battery you used to power the Windmill and the Water Wheel. On the same Island you pick up the poison is a pile of dirt (on the left side). Plant the 3rd and final seed here to grow a leaf. Cross the leaf and place the Green Battery in the empty cell to reach the Memory.

Memory VII: Freedom Isles - Symphonic Lobe
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When you have to turn the mirrors to free the instrument, you’ll have to go up and around a large gap while 6 plants shoot at you. When you reach the other side, there will be one lone mirror (shown in image 1). The correct direction to face the mirror to free the instrument is the right, but instead, turn the mirror so the orbs fly to the left. Let 1-2 orbs fly left just to be safe, then turn the mirror so they fly right. Head back up and around, then make your way to the instruments. Exhaust any dialogue and go left past the tree, up a small ramp. There is a Windmill with a wheel just to the right of it (the 2nd image shown - the wheel was previously covered with flowers but the orbs flying left cleared it). Turn the wheel to clear the smoke and grab the Memory. 

Memory VIII: Freedom Isles - Overpass of Originality
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You'll need two Green Batteries for this one. The first you can pick up off the green windmill in Freedom Promenade (img 1) and the second you can pick up after defeating the Tentacle in the Gardens of Creativity (the one used to power the green windmill and adjacent pipe - shown in img 2). When you have both, head right of the Trumpet bridge in the Overpass of Originality and use the windmills to clear the right most path. Go in this new area and place the batteries on the empty cells. A bridge will lower. Cross it to pick up the Memory.

Memory IX: Clockwork Town - Clockwork Junction

It’s recommended you wait until after you defeat the Nightmare in Clockwork Town before getting this, as you’ll have picked up the Orange Battery required for Memory X (which can be used here as well) and you won’t have to backtrack too far to get this one. When you’ve returned to Cerebrum City, turn around and head back to Clockwork Town. At the first fork, go left and up the pipe. At the top, you’ll see an empty cell next to a pipe house. Place the Orange Battery and knock on the door to get the Memory.

Memory X: Clockwork Town - Enigma Hill
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You’ll need a Blue and Yellow battery for this one. The Blue Battery can be picked up in Clockwork Junction after you have straightened out the pipes (show in img 1 - it’s the one the raised the gears, which you no longer need). The Yellow Battery can be picked up in Enigma Hill (Shown in img 2). Head right, across all the clocks and you’ll come to a pipe. At the end of the pipe are cells for the batteries (Img 3). Place each of them and a pathway will appear (Note you will lose your Blue Battery here). Go across and pick up the Memory. Be sure to pick the Yellow Battery back up once you’re done!

Memory XI: Clockwork Town - Planning Plateau

When you need to get the gears spinning on the large gear house (to lower the Heavy Thought onto the Train of Thought), don’t leave just yet. On each of the 4 buttons is a light bulb. You need to hit the buttons so all 4 buttons are lit up at the same time. Number the buttons from left to right, 1-2-3-4. Hit them in the following order: 4 - 2 - 3 - 1.

Memory XII: Clockwork Town - Composition Road

After you have rebuilt the bridge via playing the Organ, you’ll give the Train of Thought a kick forward. Before following him, head down the pipe to the right. Get all the purple squares to light up to make a platform at the end appear, allowing you to pick up the Memory.

Memory XIII: Clockwork Town - Composition Road
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When you lower the scale in Clockwork Junction, follow the Train of Thought into Composition Road but beforeyou enter, make a note of the rock on the left with the red drawing on it (Img 1). On the face of the rock are some piano keys. Remember their pattern (Down - Up - Up - Down - Up - Up - Down). In Composition Road is a large organ. Spin the wheel on top of it so the red drawing is showing (Img 2). Step on only the piano keys that were down on the face of the rock you saw earlier (only step on the 1st, 4th, and 7th keys). Doing so will reveal the Memory.


Memory XIV: Clockwork Town - Geometry Square 
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You’ll need to Metal Discs to reach this memory. You can/should have picked them up on your previous travels, but just in case you missed them… One is located in Enigma Hill (Img 1 - pick it up after you kick the Train of Thought towards Planning Plateau) and the other is in Clockwork Junction (Img 2 - pick it up after you have return with the Heavy Thought). When you have them both, head on to Geometry Square and you’ll need to find 3 gears to progress through the story. On the left side is one of the gears next to the two lightbulb houses required for Memory XV (and the Turning on the Light trophy). Just to the left of the houses is a lift without a plate (Img 3). Place one of the Metal Plates down and take the lift up. Place another Metal Plate down at the next lift and take that up as well. Follow the path around and you’ll reach the Memory. (Don’t worry about picking up the plates again, you don’t need them. A pathway will appear for you to take down instead)

Memory XV: Clockwork Town - Geometry Square, directly next Memory XIV

This Memory is not required for this trophy, but is instead needed for the Turning on the Light trophy, and is the hardest to get, but not too difficult. The object of this puzzle is to get all 5 small lights on top of the houses turned on at the same time by alternating knocking on the two doors. This is the only pattern I could find (hopefully it helps). Start with the left door (Note: you want to pay attention to what the lights do, not so much what the responses from the persons inside are):

• Left Door
  • If no lights turn on (big or small), knock again (a small light on the left house will turn on. When it does, go knock on the right door).
  • If the large light on the left house turns on, go knock on the right door (a small light on the left house will light up. Knock on the right door again when it does).
• Right Door
  • If no lights turn on, go knock on the left door (a small light on the right house will light up - when it does, knock on the left house again).

Basically… Whenever a small light turns on, you always want to knock on the opposite door next. 
Whenever one of the two large lights turn on, you always want to knock on the opposite door next. It may take a few, but as soon as all 5 Lights are lit, the Memory will appear. 

Memory XVI: The Pathway

This is the only Memory found in The Pathway and is not needed for this trophy (it will unlock Waterboy). Near the beginning, you’ll come across a disheveled house with a large nail piercing the side of it. Piper will make an audible comment about the house and her and Dusty will share a small conversation regarding the previous tenant. Just to the right of the house is a dry plant. The ice cubes you picked up during the introduction have now turned to water in your inventory. Use the water on the plant to make it grow and it will release the last Memory.