When can one use a $O(n)$ time sorting algorithm?
In the comparison model, where all you are allowed to do is to compare two elements, and without further assumptions, we can prove that no sorting algorithm can do better than $O(n\log n)$.

If you want to sort in $O(n)$, you need either a stronger model, or additional assumptions.

For example, if you can bound the range of the numbers you are sorting, you can use bucket-sort, which is $O(n)$ (time).

A different example is spaghetti-sort: if you can implement the $\max$ function over $n$ elements in $O(1)$, then you can sort in $O(n)$.

You see here that different assumptions can allow you to sort in $O(n)$. There is no characterization of exactly which assumptions allow it.