Abstract: We will introduce plaque inverse limits of branched covering self-maps of simply-connected Riemann surfaces. We will define the notions of regular and irregular points. At a regular point, a plaque inverse limit has a natural Riemann surface structure, which was studied in the literature since 1990s. However, its local structure at the irregular points was not previously known and is of a great interest. We will introduce certain algebraic machinery, which will permit us to define new local invariants of plaque inverse limits. These invariants, which we call signatures, are trivial at, and only at, regular points. At irregular points the signatures are closely related to the dynamics of the iterations. We will correlate local topological properties of a plaque inverse limit at a point and its signatures. Finally, we will present several examples which are of interest in Holomorphic dynamics. Using dynamical properties, we will construct irregular points and compute their signatures. A conjecture will be presented, which, if true, permits us to build a counter-example to the reverse direction of Mane's Theorem.