Ordinal-definable subsets of singular cardinals

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HOD _x contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ ⁺ is supercompact in HOD _x for all x ⊆ κ.