I am not familiar with the current studies on the Hausdorff dimension of the singular set of solutions to PDE, but I know that in [1] a necessary and sufficient condition for the holding of Hartogs phenomenon for a linear system of partial differential operators was stated and proved. The author does not use the methods of geometric measure theory: however, he deals with general compact singularities for the solutions, including the ones with non-zero Lebesgue measure.
[1] Gaetano Fichera (1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali [Hartogs phenomenon for certain linear partial differential operators]" (Italian), Rendiconti dell' Istituto Lombardo di Scienze e Lettere. Scienze Matematiche e Applicazioni, Series A., 117: 199–211, MR0848259, Zbl 0603.35013.