= Solution
On a maximal torus element $\operatorname{diag}(t,t^{-1})$, the <SU(2) representation> $V_2$ has weights $2,0,-2$. A monomial with respective multiplicities $a,b,c$ has weight $2(a-c)$, giving
$$
\boxed{\chi_{\operatorname{Sym}^n(V_2)}(t)=\sum_{a+b+c=n}t^{2(a-c)}.}
$$
For $0\leq\ell\leq n$, its weight-$2\ell$ multiplicity is $\lfloor(n-\ell)/2\rfloor+1$, and multiplicities are symmetric in $\ell$. An irreducible $V_{2j}$ contributes one at each weight $2j,2j-2,\ldots,-2j$, so subtracting adjacent multiplicities yields the <symmetric powers of the three-dimensional SU2 representation> decomposition
$$
\operatorname{Sym}^n(V_2)\cong\bigoplus_{j=0}^{\lfloor n/2\rfloor}V_{2n-4j}.
$$
It contains the trivial representation once when $n$ is even, and never when $n$ is odd. Therefore
$$
\boxed{\dim\operatorname{Sym}^n(V_2)^{SU(2)}=\begin{cases}1,&n\text{ even},\\0,&n\text{ odd}.\end{cases}}
$$
The double cover $SU(2)\to SO(3)$ identifies $V_2$ with the complexified standard three-dimensional representation. It is self-dual, so the same invariant dimensions apply to homogeneous polynomials. The quadratic form $q=x^2+y^2+z^2$ is invariant, and its powers give nonzero invariants in every even degree. Since each such degree has dimension one and each odd degree has dimension zero, every invariant is a polynomial in $q$. There is no polynomial relation on $q$, as seen by setting $y=z=0$. Thus
$$
\boxed{\mathbb C[x,y,z]^{SO(3)}=\mathbb C[x^2+y^2+z^2].}
$$
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