Has there ever been an application of dividing by $0$?

You said you wanted an application. Inspired by the example from Exceptional Floating Point, consider the parallel resistance formula:$$ R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} $$This formula tells you the effective electrical resistance of a path when the current can choose two routes to take.

Let's pretend that $R_1=0$. Then we have:$$ R_{total}=\frac{1}{\frac{1}{0}+\frac{1}{R_2}}=\frac{1}{\infty+\frac{1}{R_2}}=\frac{1}{\infty}=0 $$The resistance being zero is indeed the correct answer; all current flows along the single wire that has no resistance.

Naturally, you need to make appropriate definitions for arithmetic on $\infty$ (i.e., use the projective reals). For well-behaved applications like this, that's fairly straightforward.

In complex analysis, we talk about the value of a function at infinity. To evaluate $f(z)$ at infinity, compute $f(1/z)$ then plug in $0$. This allows us to talk about things like the order of zeros and poles at infinity.

Example: $$f(z) = \frac{az+b}{cz+d}$$ with $ad-bc \neq 0$ and say $a,c \neq 0$. $$f(1/z) = \frac{\frac{a}{z}+b}{\frac{c}{z}+d} = \frac{a+bz}{c+dz}$$ So $f(\infty) = \frac{a}{c}$.