Video solution:

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Written solution:

Since the sum is $0,$ all the numbers cannot be positive or negative. This gives two cases: two are positive and one is negative or vice versa. Also note that if $x < 0,$ $\dfrac{x}{|x|} = -1,$ and if $x > 0,$ $\dfrac{x}{|x|} = 1.$

$\textbf{Case I:}$ two are positive and one is negative

WLOG, let $a, b > 0$ and $c < 0.$ Then $abc < 0.$ This means that $\dfrac{a}{|a|} = \dfrac{b}{|b|} = 1$ and $\dfrac{c}{|c|} = \dfrac{abc}{|abc|} = -1.$ Then $$\dfrac{a}{|a|}+\dfrac{b}{|b|}+\dfrac{c}{|c|}+\dfrac{abc}{|abc|} = 0.$$

$\textbf{Case II:}$ two are negative and one is positive

WLOG, let $a, b < 0$ and $c > 0.$ Then $abc > 0.$ This means that $\dfrac{a}{|a|} = \dfrac{b}{|b|} = -1$ and $\dfrac{c}{|c|} = \dfrac{abc}{|abc|} = 1.$ Then $$\dfrac{a}{|a|}+\dfrac{b}{|b|}+\dfrac{c}{|c|}+\dfrac{abc}{|abc|} = 0.$$

Either way, $\dfrac{a}{|a|}+\dfrac{b}{|b|}+\dfrac{c}{|c|}+\dfrac{abc}{|abc|}$ equals $0.$

Thus, A is the correct answer.