Are $AC^{0}$ algorithms interesting to research communities?
$\mathsf{AC^0}$ algorithms are quite weak. For example, $\mathsf{AC^0}$ cannot compute parity or majority. However, it can compute approximate majority, as shown by Ajtai; see Viola, On Approximate Majority and Probabilistic Time, for a better construction and for references to Ajtai's work.

$\mathsf{TC^0}$ is a slightly stronger class, also consisting of polynomial size circuits having constant depth, but allowing threshold gates. Such circuits can compute division, as shown by Hesse, Division is in Uniform $\mathsf{TC^0}$, improving on works of others (starting with Beame, Cook and Hoover). The related problem of iterated integer multiplication is also in $\mathsf{TC^0}$.