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We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups 
$\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for 
$\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.
