Rewrite the metric as follows:

$$ds^2 = V (dx + A)^2 + \frac1V (dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2)$$

This exhibits the Taub-NUT metric as the metric on the total space of a circle bundle over $\mathbb{R}^2\setminus\{0\}$. Let

$$\vartheta_1 = \sqrt{V} (dx + A) ~~~~~~~~ \vartheta_2 = \frac1{\sqrt V} dr$$

$$\vartheta_3 = \frac1{\sqrt V} r d\theta ~~~~~~~~ \vartheta_4 = \frac1{\sqrt V} r \sin\theta d\phi$$

so that we may write the metric as

$$ ds^2 = \vartheta_1^2 + \vartheta_2^2 + \vartheta_3^2 + \vartheta_4^2$$

and the volume form is

$$d\mathrm{vol} = \vartheta_1 \wedge \vartheta_2 \wedge \vartheta_3 \wedge \vartheta_4 = \frac{r^2\sin\theta}{V} dx \wedge dr \wedge d\theta \wedge d\phi$$

If $\theta = \pi$, then $\sin\theta = 0$ and hence $d\mathrm{vol}=0$, whence it is singular.