According to some presentations, understanding the refutation of sakkāyadiṭṭhi can come in many forms from the coarsest to the subtlest.
The coarsest level is linking sakkāyadiṭṭhi to the belief in a permanent, unitary and independent thing. Permanence here refers to not arising and not disintegrating; unitary means being a single entity without component parts; and independence means not depending on conditions. Understanding that no thing can exist in such a manner is the coarsest level of refutation of sakkāyadiṭṭhi.
Believing things exist counter to this coarsest level of refutation comes from acquiring intellectual/conceptual beliefs about how things exist. We acquire this mistaken belief from contact and learned familiarity with a system of philosophy or mode of thinking that adopts this view.
Understanding impermanence itself can come in coarse/gross and subtle forms. Gross impermanence can be seen with the obvious perception of how constructed things eventually fall into ruins. A constructed building or shelter ages naturally and eventually crumbles given enough time. This is gross impermanence.
Subtle impermanence is harder to see; it is the understanding that from the moment any thing is constructed there is not even an instant of static unchanging duration. All things, from the moment of arising, are in a constant flux of change; this is their subtle impermanence. This is important to understand because it follows that a thing’s eventual demise inevitably follows from its very construction. The cause of construction is also the cause of destruction.
Independence can also be understood in coarser and subtler ways. There is the coarse version which just means not being affected by or relying on causes and conditions for arising and disintegrating. The subtler version means able to exist or be posited in a self-sustaining and independent manner without relying upon anything else. An independent thing in this sense would not rely upon anything relative to itself in order to be posited. An even deeper and subtler understanding comes from realizing that nothing can exist without relying upon being imputed depending on name and concept.
In terms of the person or self, refuting sakkāyadiṭṭhi fully entails all of the coarse and subtle understandings above. It also entails realizing that for a person to have substantial existence it would mean that they could be known to exist independent of the aggregates. That is, the person could be known without the aggregates appearing to the mind. However, the person cannot be known in this way. Rather, the person is merely imputed to exist based on the appearance of the aggregates to the mind. Ordinary persons constantly mistake the appearance of the aggregates for a substantially existent person. This is not an acquired mistake, but rather has been with us for an unfathomable amount of time.
This latter non-acquired mistaken sakkāyadiṭṭhi does not depend upon reason or intellectual concepts and therefore can not be overturned and refuted through reason or logic. It also does not depend upon the view of a person or self, but rather is a subtle ignorance about how all things exist. Without overturning this very subtle ignorance of substantial existence with regard to all things, the soteriological goal of the path cannot be fully realized.
All the above is my very poor understanding of what my teacher’s have tried to pound into this very limited and thick skull and all errors or mistaken beliefs in the above are mine alone. No doubt I’ve got lots wrong and stated the situation quite ineloquently. Much of it comes from a very poor paraphrase of the book, Insight into Emptiness by Khensur Jampa Tegchok. 
