I am aware that at least for lower dimensions, 

> "smooth manifolds iff triangulable manifolds" 
>
> at least for dimensions below a certain critical dimensions D.

My question is that for 

>- For orientable manifolds, in which lower dimensions $\leq$ D, such that 
"smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.") 

Consider:

1. orientable SO-manifolds with SO(D) co/bordism structure.

2. orientable Spin-manifolds with Spin(D) co/bordism structure.


>- For non-orientable manifolds, in which lower dimensions $\leq$ D, such that "smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.") 

Consider:

3. non-orientable O-manifolds with O(D) co/bordism structure.

What are their critical dimensions D?

In [this post](https://mathoverflow.net/q/264330/27004), we learn: 

>"All orientable 5-dimensional manifolds are triangulable." 
>"In 6 dimensions, there are non-triangulable orientable manifolds."
>
>Are these referred to topological manifolds? Or smooth manifolds?


However, I heard an alternative view from a geometry topologist that (maybe standard viewpoint to some of you, but I apologize for my background ignorance): 

> "All smooth manifolds are uniquely triangulable.  No critical dimensions D constraint or orientability constraints."




p.s. Your correct answers are more important. References (the proofs) are secondary but very helpful.


Many thanks! Really appreciate your holiday time answer and inspiration!