We consider two kinds of gauge-theoretic equations; the Vafa-Witten equations and Kapustin-Witten ones on closed 4-manifolds, which can be seen as generalizations of the Hitchin equtions on Riemann surfaces. One of shared problems in the study of solutions to these equations is unboundedness of them especially in the direction of the Higgs fields. Regarding this, Cliff Taubes rather recently made a wonderful breakthrough by looking at a real codimension two singular set of the underlying 4-manifold for a sequence of solutions to the equations so that the corresponding sequence of "rescaled" Higgs fields (and connections as well in the Kapustin-Witten case) has a converging subsequence outside it and a finite set of points corresponding to "bubbling" after gauge transformations. So one aspect of the problem can be looked into by sorting out what these singular sets would look like.
This talk reveals it in some situations. Firstly, we see that the singular set is empty under certain no bubbling condition. This leads to an observation that relates the irreducibility of the connections with the boundedness of the Higgs fields in the equations. We then consider the equations on a compact Kahler surface X, and figure out the singular sets in this case have the structure of analytic subvariety of X.