We consider Gelfand pairs
(
G
,
K
)
(G,K)
where
G
G
is a compact Lie group and
K
K
a subgroup of fixed points of an involutive automorphism. We study the regularity of
K
K
-bi-invariant matrix coefficients of unitary representations of
G
G
. The results rely on the analysis of the spherical functions of the Gelfand pair
(
G
,
K
)
(G,K)
. When the symmetric space
G
/
K
G/K
is of rank
1
1
or isomorphic to a Lie group, we find the optimal regularity of
K
K
-bi-invariant matrix coefficients of unitary representations. Furthermore, in rank
1
1
we also find the optimal regularity of
K
K
-bi-invariant Herz-Schur multipliers of
S
p
(
L
2
(
G
)
)
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
K
-bi-invariant matrix coefficients of unitary representations of
G
G
are
α
\alpha
-Hölder continuous for some
α
>
0
\alpha >0
, then all
K
K
-finite matrix coefficients of unitary representations are also
α
\alpha
-Hölder continuous.