Singular continuous measures "in nature"

Solution 1:

Any transformation that makes you loose some dimension will transform a continuous measure into a type three measure. For example if you have a measure in $\mathbb R^n$, the measure inferred by the transformation $\mathbf x \rightarrow \frac{\mathbf x}{\lVert \mathbf x \rVert}$ will have a support that is Lesbegue-null in $\mathbb R^n$. The same will happen with any linear transformation from $\mathbb R^n$ to $\mathbb R^n$ that is not onto (the $n\times n$ transformation matrix is not full rank).

These scenario seems kind of cheating about the problem but I think that what is bothering you is that the Cantor does not have integer dimension which makes it less intuitive to us.