= Solution
The spin components satisfy
$$
[S_i,S_j]=i\hbar\varepsilon_{ijk}S_k,
\qquad
[S^2,S_i]=0.
$$
The irreducible representations are indexed by
$$
s=0,\frac12,1,\frac32,\ldots
$$
and have orthonormal basis $|s,\sigma\rangle$, where
$$
\sigma=-s,-s+1,\ldots,s.
$$
On this basis,
$$
S^2|s,\sigma\rangle=\hbar^2s(s+1)|s,\sigma\rangle,
\qquad
S_z|s,\sigma\rangle=\hbar\sigma|s,\sigma\rangle.
$$
Each Cartesian component is Hermitian,
$$
S_i^\dagger=S_i,
$$
and $S_+^\dagger=S_-$. Also
$$
\operatorname{tr}S_z=\hbar\sum_{\sigma=-s}^s\sigma=0.
$$
Rotational equivalence, or the off-diagonal form of $S_x,S_y$ in this basis, gives
$$
\boxed{\operatorname{tr}S_x=operatorname{tr}S_y=operatorname{tr}S_z=0.}
$$
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