Growth rate of $n^{\sin n}$

Neither of these functions are $O$ or $\Omega$ of the other, since neither $f(n)/g(n)$ nor $g(n)/f(n)$ is bounded.

In other words, they are not asymptotically comparable.

There are infinitely many integers $n$ such that $\sin n\gt\frac{\sqrt{3}}{2}\gt \frac{1}{2}$.

For note that $n+1$ differs from $n$ by $1$ radian, a bit under $60$ degrees. Thus infinitely many $n$ fall between $2\pi k+\frac{\pi}{3}$ and $2\pi k+\frac{2\pi}{3}$, where $k$ ranges over the positive integers. In that interval, the sine function is greater than $\frac{\sqrt{3}}{2}$.

Added: The above shows that for any positive constant $c$, there are (infinitely many) $n$ such that $n^{\sin n}\gt cn^{1/2}$.

This is because for infinitely many $n$, we have $\frac{f(n)}{g(n)}\gt n^{(1/2)(\sqrt{3}-1)}$.

Thus $n^{\sin n}$ is not $O(n^{1/2})$.

The other direction is easy, $n^{\sin n}\lt 1$ for infinitely many $n$, so $n^{1/2}$ is not $O(n^{\sin n})$.