Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent with a measurable cardinal $\kappa$ satisfying $o(\kappa)=\kappa^{++}$.

But this failure is at $\aleph_\omega$. Suppose we wanted more.

Suppose that we wanted the failure to happen on a couple isolated points. Well, it's not hard to redo the standard constructions and get just that. But what happens when we have limit points?

Even more, by Silver's theorem if SCH fails at $\kappa>\operatorname{cf}(\kappa)>\omega$, then there is a stationary subset of $\kappa$ where SCH failed.

What would be the consistency strength when $\kappa$ is a singular limit of singular cardinals, and SCH fails cofinally below $\kappa$? What if we require $\kappa$ to be of uncountable cofinality?

As a side question, what if $\kappa$, with uncountable cofinality, does satisfy SCH, but an unbounded subset (which has to be non-stationary, of course) of it does not?