Here is my best effort to describe the all the different possible scenarios and the chance that they will turn a Shiny Pokemon.
I've come across numerous different conflicting reports on the exact chances, especially for Masuda method. I went with my best understanding of these mechanics, but have had to trust the accuracy of Bulbapedia (which conflicts itself) and Serebii (which presents confusing info) for some of these figures.
Many sources mention 1/683 for certain methods, but this is a slightly inaccurate conversion of 3/2048, which comes from 6/4096 which comes from the five extra personality values** being generated for the encounter using certain methods. Shiny Charm adds another two, bringing the chance to 8/4096 or 1/512 for one of the methods. Hopefully that makes sense.
- Standard encounter or egg = 1/4096
- Standard encounter or egg with Shiny Charm = 3/4096
- SOS encounter, 70+ chains = 1/1024*
- SOS encounter, 70+ chains with Shiny Charm = 3/2048* (3/1024?)
- Masuda method breeding = 3/2048**
- Masuda method breeding with Shiny Charm = 1/512
*These are chances claimed by Serebii. Its number, 1/1024, suggests the chances are multiplied by 4 when on the 70th chain. However, it then goes on to state that Shiny Charm increases this to 1/683 (rather, 3/2048 as explained above), which doesn't make any sense because that number happens when five personality values are added to the original 1/4096 chance, which is not applicable if the chance is instead 1/1024 and we're using Shiny Charm. So the chance should be 3/1024 if I'm not mistaken. 3/2048 might actually make sense if the original chance for reaching a 70 chain is 1/2048, not 1/1024.
**This information comes from Bulbapedia, which I have come to distrust with regards to mechanics like these, though it is still one of the best sources of info available. The article I used first claims that Masuda method multiplies the chance by 4, but then later on states that it actually adds five personality values making the chance 6/4096 (again, 3/2048 rounded not 1/683). I trust the latter explanation more, because it makes sense with the chances described by other sources I looked at, which mention 1/512 which makes sense if it comes from 8/4096, which is what you'd get from the five extra personality values plus the other two from Shiny Charm. This particular explanation also originated from research by Smogon, which I trust.
Sorry if any of that was confusing. The internet is unfortunately very indecisive when it comes to these mechanics.
Source, source, source, source