Diverges by the Divergence Test since $\lim_{n \to \infty} \sin(n) \neq 0$

Converges by Dirichlet's Test (partial sums of $\sin(n)$ are bounded, and $1/n \to 0$ monotonically)

Converges absolutely by Comparison Test with $1/n^2$

Diverges by Integral Test since $\int_1^{\infty} \sin(x)/x \, dx = \infty$