I'll try to break down a list with some favorite techniques I like to use and that you can use to look up some tutorials.
I would like to refer to my 2019 #popuptober post (Youtube playlist link) where I joined the challenge to make one pop-up a day based on a word, for 31 days long. - I've used numbers for the days that I can refer to below.
1 Parallel fold (06, 09, 14, 26)
The most common and classic technique that creates layers and extra angles to work with. It can be used in 90 and 180 degrees (opened) pop-ups. In a 90 degree design, it can be used to add layers to a scene while used in a 180 degrees pop-up, it creates an extended fold that moves along with the centerfold.
2 V-fold (06, 19, 22, 24)
V-folds create a lot of motion and are named after their V shape with a fold in the center. One side of the V-fold can be used as a rotating surface. Together they create and upwards motion that can be used to lift up a surface or piece.
3 Asymmetric V-fold (05, 12, 17)
The asymmetric V-fold creates a horizontal upward movement. This can be used for all kinds of scenes that need this animation or movement (animation is used as a term to explain movement in pop-ups).
4 V-fold twister/rotator (no examples in the popuptober posts)
When you place two V-folds on the centerfold of the page with the points almost touching, you have the base of a V-fold twister. When you connect the opposite halves of both V-folds with a "bridge" you'll get a mechanism that rotates. This base can be extended with other mechanisms and added platforms.
5 Diagonal box (04, 27)
A box shape that gets lifted by two sides. A perfect mechanism to expand with added mechanisms like parallel platforms. It's strong and can be pushed into an almost perfect cube shape.
6 Straight box (09, 20)
A box that provides a solid base with many flats. It's less square than a diagonal box because of the two extra folds that are needed to make it fold in and out. But very useful to build up a structure or high platforms.
7 (parallel) Platforms (02, 14, 10)
This is a platform that doesn't move and has a top that makes the same movement as the centerfold it's placed on. This technique is used a lot in pop-ups that have layers or tall structures. Without securing the structure it can be moved (also on purpose ref: coffin dance card) and act wobbly. It can be "locked" by adding extra horizontal support.
8 "The boat" (02, 11, 15, 17, 23, 25)
I'm adding "" to this technique's name because I like to explain it more as a way to pull out a structure. I also use this technique to create curved shapes. By attaching two ends of a 3D structure to the page (with the fold centered), it gets pulled into shape when you open the page. I've built a lot of real boats with this technique but also use it to create landscapes and structures. It's strong and very useful.
9 V-fold raisers (01, 07, 15, 23, 28, 31)
V-fold raisers can be very useful and effective for making upward movements. It's basically two V-folds that provide a structure to attach a platform to. By making the V-folds taller, the upward movement will also increase but can make the structure weaker.
10 Pull tab transformations (03, 13, 16)
This may be one of my favorite techniques but I've saved this for last because it's actually a combination of different mechanisms. The basic idea is that you pull a strip and the page or an element on the page transforms. It's fun because it creates interaction.