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The notion of operator amenability was introduced by Z.-J. Ruan in 1995. He showed that a locally compact group $G$ is amenable if and only if its Fourier algebra $A(G)$ is operator amenable. In this work we investigate the operator amenability of the Fourier-Stieltjes algebra $B(G)$ and of the reduced Fourier-Stieltjes algebra $B_r(G)$. The natural conjecture is that any of these algebras is operator amenable if and only if $G$ is compact. We partially prove this conjecture with mere operator amenability replaced by operator $C$-amenability for some constant $C< 5$. In the process, we obtain a new decomposition of $B(G)$, which can be interpreted as the non-commutative counterpart of the decomposition of $M(G)$ into the discrete and the continuous measures. We further introduce a variant of operator amenability - called operator Connes-amenability - which also takes the dual space structure on $B(G)$ and $B_r(G)$ into account. We show that $B_r(G)$ is operator Connes-amenable if and only if $G$ is amenable. Surprisingly, $B(F_2)$ is operator Connes-amenable although $F_2$, the free group in two generators, fails to be amenable.