  We consider symmetric Gelfand pairs $(G,K)$ where $G$ is a compact Lie group
and $K$ a subgroup of fixed point of an involutive automorphism. We study the
regularity of $K$-bi-invariant matrix coefficients of $G$. The results rely on
the analysis of the spherical functions of the Gelfand pair $(G,K)$. When the
symmetric space $G/K$ is of rank $1$ or isomorphic to a Lie group, we find the
optimal regularity of $K$-bi-invariant matrix coefficients. Furthermore, in
rank $1$ we also show the optimal regularity of $K$-bi-invariant Herz-Schur
multipliers of $S_p(L^2(G))$. We also give a lower bound for the optimal
regularity in some families of higher rank symmetric spaces. From these
results, we make a conjecture in the general case involving the root system of
the symmetric space. Finally, we prove that if all $K$-bi-invariant matrix
coefficients of $G$ have the same regularity, then so do all $K$-finite matrix
coefficients.
