The Lord of the Rings: Gollum

There are 35 total collectibles in the game, spread out across all ten chapters. None are missable, since you can use Chapter Select from the main menu to mop up any you've missed. Also from the main menu is a Collectibles option where you can see all the ones you have, and cross reference it with the list below to see which you're missing. Unfortunately, there's no other way to easily check what you're missing from where. Here is a list of what is in each chapter, with a video guide at the end.

Chapter 1

1. Whetstone

Chapter 2

1. Boroc Hair
2. Bat Wing
3. Piece of Leather
4. Piece of Metal
5. Old Metal Spoon

Chapter 3

1. Feather

Chapter 4
None

Chapter 5

1. Picture of Baggins
2. Eye of Spider
3. Piece of Boroc Tusk
4. Nail
5. Piece of Metal
6. Dead Beetle

Chapter 6

1. Orc Tooth

Chapter 7

1. Rope
2. Fish Bones
3. Shell
4. Brooch
5. Animal Figurine
6. Piece of Marble
7. Splinter
8. Shiny Button
9. Piece of Marble
10. Harp String
11. Pearl
12. Badger's Claw
13. Twig and Flower
14. Bird Bones

Chapter 8

1. Elven Figurine
2. Shard

Chapter 9

1. Wooden Fork
2. Piece of Garland

Chapter 10

1. Little Bell
2. Nut
3. Hook
4. Piece of a Mirror

Here is a video guide showing the locations of all 35 collectibles. Full credit to the creator of the video.