  Regular nilpotent Hessenberg varieties form an important family of
subvarieties of the flag variety, which are often singular and sometimes not
normal varieties. Like Schubert varieties, they contain distinguished points
called permutation flags. In this paper, we give a combinatorial
characterization for a permutation flag of a regular nilpotent Hessenberg
variety to be a singular point. We also apply this result to characterize
regular nilpotent Hessenberg varieties which are normal algebraic varieties.
