I‘ve seen Leibniz’s, Lagrange’s, Euler’s and Newton’s notation for derivatives. They’re quite different, and I suppose they all have different applications where they shine the brightest.
In what circumstances are the different notations most common? Is there a reason to have multiple notations for the same thing?
When a concept is as important as the derivative is, and with as complex a history, and used in as many niche applications, it is prone to having many different notations that are used within different circles. The short answer is, whatever area you're working in, see what notation textbooks/professors/papers in the field typically use. If everyone in your field uses the same notation, use that one.
Here are a few loose guidelines from my experience:
The notation $\dot y$ is typically used to mean the derivative with respect to time. It is very common in classical mechanics.
The notation $dy/dx$ is typically used more in applied math and the notation $f'(x)$ is typically used more in pure math. This one is very rough and should be taken with a large dose of salt.
The partial derivative notation $\partial f/\partial x$ is the "default" and the most common. It is also the most classical.
In partial differential equations, the notation $f_x$ for the partial derivative w.r.t. $x$ is very common.
I think mathematicians working with differential forms seem to like $D_x f$ for the partial derivative w.r.t. $x$. Many feel that $\partial f/\partial x$ is just too clunky, especially when written in matrices.
Sometimes other notations will appear. Once, in a very specific area I was working in, the notation was $\delta y/\delta t$!