It is consistent that there exists a singular cardinal κ of uncountable cofinality such that 2^κ is weakly inaccessible and every regular cardinal strictly between κ and 2^κ is the character of some uniform ultrafilter on κ.
Merimovich, Extender based Magidor-Radin forcing , Israel Journal of Mathematics 182 (2011), 439–480
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Ultrafilters on singular cardinals of uncountable cofinality
It is consistent that there exists a singular cardinal κ of uncountable cofinality such that 2^κ is weakly inaccessible and every regular cardinal strictly between κ and 2^κ is the character of some uniform ultrafilter on κ.