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Abstract:  In this talk,  I will present  a singular version of the Sphere Covering Inequality  and   its applications  in the study  of singular Liouville-type problems with super harmonic weights.  In particular,  new uniqueness  results for solutions of the singular mean field equation both on spheres and on bounded domains will be discussed.  This approach  also gives a new self-contained proofs of  previously known results, such as the uniqueness of spherical convex polytopes first established by Luo and Tian.   Furthermore,  I will also  show  new symmetry results for the spherical Onsager vortex equation.