  We establish universal Gaussian fluctuations for the mesoscopic linear
eigenvalue statistics in the vicinity of the cusp-like singularities of the
limiting spectral density for Wigner-type random matrices. Prior to this work,
the linear eigenvalue statistics at the cusp-like singularities were not
studied in any ensemble. Our analysis covers not only the exact cusps but the
entire transitionary regime from the square-root singularity at a regular edge
through the sharp cusp to the bulk. We identify a new one-parameter family of
functionals that govern the limiting bias and variance, continuously
interpolating between the previously known formulas in the bulk and at a
regular edge. Since cusps are the only possible singularities besides the
regular edges, our result gives a complete description of the linear eigenvalue
statistics in all regimes.
