When is $0^{0}=0$ useful? Are there mathematical areas/situations in which defining $0^{0}$ as $0$ is useful/sensible and convention (in contrast to the "common" definition as $1$) ?
 A: The $p$-norm is a generalization of the standard Euclidean norm for vectors. It is defined as:
$${\parallel\mathbf{x}\parallel}_p = \left(\sum_{i = 1}^{n} \left|x_i\right|^p\right)^{1/p}$$
This is for $p > 0$. What about $p = 0$?
Well, maybe we can call that the $0$-norm. The $0$-norm, is very useful in measuring the "sparsity" of a vector. A vector is $s$-sparse if only $s$ of its components are non-zero. It's useful to have $0^0$ be defined as $0$, while any other $r^0$ (where $r \neq 0$) is $1$. Okay, so the $0$ norm is not quite a $p$-norm (because we drop the $1/p$, but it is analogous. Actually, the zero norm is not even a norm (it's not scalable). Oops.
Anyway, so the $\ell_0$-norm (the common name for this measurement), for measuring the sparsity of a vector, looks something like:
$$ {\parallel\mathbf{x}\parallel}_0 = \left(\sum_{i = 1}^{n} \left|x_i\right|^0\right)$$
Why is sparsity a big deal? Well, compressed sensing is one good reason why. 
Basically, compressed sensing tells us that under certain conditions (the signal is sparse, and the sensing matrix is randomly distributed), we can reconstruct a signal with even less information than is required by Nyquist-Shannon theorem. 
