Step 1: Note the given information.
We are given $\sin\theta = -\dfrac{3}{4}$ and we need to find $\sin 2\theta$.
Step 2: Use the double angle formula.
$\sin 2\theta = 2\sin\theta\cos\theta$. We already know $\sin\theta = -\dfrac{3}{4}$, so we need $\cos\theta$.
Step 3: Find $\cos\theta$ using the Pythagorean identity.
$\cos^2\theta = 1 - \sin^2\theta = 1 - \dfrac{9}{16} = \dfrac{7}{16}$. So $\cos\theta = \pm\dfrac{\sqrt{7}}{4}$.
Step 4: Compute $\sin 2\theta$ for $\cos\theta = \dfrac{\sqrt{7}}{4}$.
$\sin 2\theta = 2 \cdot \left(-\dfrac{3}{4}\right) \cdot \dfrac{\sqrt{7}}{4} = -\dfrac{6\sqrt{7}}{16} = -\dfrac{3\sqrt{7}}{8}$.
Step 5: Compute for the other case $\cos\theta = -\dfrac{\sqrt{7}}{4}$.
$\sin 2\theta = 2 \cdot \left(-\dfrac{3}{4}\right) \cdot \left(-\dfrac{\sqrt{7}}{4}\right) = +\dfrac{3\sqrt{7}}{8}$. The answer matching the options is $-\dfrac{3\sqrt{7}}{8}$ (when $\theta$ is in the third or fourth quadrant with $\sin\theta < 0$ and $\cos\theta > 0$).
Step 6: State the final answer.
$\sin 2\theta = -\dfrac{3\sqrt{7}}{8}$. \[ \boxed{-\dfrac{3\sqrt{7}}{8}} \]