When are tris used effectively? When are tris appropriate to use on a model and where are they usually seen?

For example, this image of a popcorn container (from turbosquid), uses tris for the top half circles and the edge corners on the bottom. Is this use of tris appropriate for this kind of model? 

Why are triangles added at the bottom corners?
In terms of quality I suggest you to model in quads. The simplest reason is:


  The flexibility to soften and sharpen edges is key to build high quality
  models.




Example geometry with a soft edge on the left side and more sharp edge on the right  

The best way to achieve this, is to build consistent topology to be fast in selecting  Alt+Shift+RMB and creating edge loops Ctrl+R, which depends on quads.



Create edge loop issue 

For the picture of your question, I assume that automatic algorithms (e.g. bevel modifiers) have been applied to improve the edges. A side effect of such algorithms is that unwanted edges appear (triangles at the corners), but I can't really see if there is a reason for this. 


  For sure there are exceptions to use triangles.


In terms of speed artists often use modifiers on simple geometry to build complex shapes - side effect can be a geometry of triangles. For the crumpled paper I could also imagine that here some kind of modifier on a icosphere was used.

But it really depends on what do you want to achieve.