Analysis

This question mainly examines the relationship between the inclination angle and the slope of a line, testing the student's calculation ability. It is quite basic.

Solution

Let the inclination angle of the line $x\sin \theta +\sqrt{3}y+2=0$ be $\alpha$,
then $\tan \alpha= -\frac{\sin \theta}{\sqrt{3}} \in \left[-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right]$

Therefore, $\alpha \in \left[0, \frac{\pi}{6}\right]∪\left[ \frac{5\pi}{6},\pi\right)$,
Hence, the correct choice is $\boxed{C}$.