  In this paper, we shall discuss possible theories of defining equivariant
singular Bott-Chern classes and corresponding uniqueness property. By adding a
natural axiomatic characterization to the usual ones of equivariant Bott-Chern
secondary characteristic classes, we will see that the construction of Bismut's
equivariant Bott-Chern singular currents provides a unique way to define a
theory of equivariant singular Bott-Chern classes. This generalizes J. I.
Burgos Gil and R. Li\c{t}canu's discussion to the equivariant case. As a
byproduct of this study, we shall prove a concentration formula which can be
used to prove an arithmetic concentration theorem in Arakelov geometry.
