Mathematical Definition of Point Source Wikipedia describes a mathematical definition of a point source as "a singularity from which flux or flow is emanating".
The usual definition in Physics describes it just as a source whose dimensions are negligible in comparison to another variable you're relating it to, which leads to well-behaved phenomena such as isotropic emission.
Well, if there's something I wouldn't expect from a singularity, that's well-behaving. Given that, how does that math definition of point source makes sense? How to phenomenologically relate it, lucidly, it with physics?
 A: A point source is described by its math, period. And since you can meaningfully calculate fields from a charge distribution that is infinitely tall at a single point (which is represented by a mathematical distribution, aka delta-function), there is nothing more to that. It is an idealization, and may or may not represent nature accurately. Eg. electrons are considered point-like particles today, while protons are not. But we can describe both at least by approximating them as point-sources in certain contexts.
A: None of the objects of mathematical geometry actually exist in the real world. However, they are useful as models of things in the real world. Use a point as a model when the physical object is small enough that its size isn't an important parameter of the problem.
A: "Point" sources are typically represented by $\delta$-functions.
For example, an electron with charge $e$ located at $\vec{r}_e$ must have charge density
$$\rho(\vec{r}) = \left\{{0,\, \vec{r}\not=\vec{r}_e\atop\infty,\, \vec{r}=\vec{r}_e}\right.,$$
and we therefore write that
$$\rho(\vec{r})=e\,\delta^{3}(\vec{r}-\vec{r}_e),$$
so
$$Q=\int_V\rho(\vec{r})\,d^3r=e,$$
as required
And note that the $\delta$-function also possesses various additional useful mathematical properties, e.g.,
$$ \nabla^2\left(\frac1{|\vec{r}-\vec{r}\,'|}\right)=-4\pi\,\delta^3(\vec{r}-\vec{r}\,'),$$
useful when evaluating typical E & M expressions involving $\nabla^2$.
