Reddit reviews Fourier Analysis on Number Fields (Graduate Texts in Mathematics (186))

A variety of places, here's some references:

Number Theory: The Mellin Transformation is connected to Dirichlet Series, in particular the Riemann Zeta function, you might try section 5.1 of Montgomery and Vaughan's Multiplicative Number Theory. There is this nice write up Fourier Analysis in Additive Number Theory or the Springer Book Fourier Analysis on Number Fields (a number field is a particular kind of extension of the rationals)

Representation Theory: Fourier Analysis on Finite Groups or Terras' lovely book Fourier Analysis on Finite Groups which has applications in Families of Expander Graphs

I confess that I'm dodging the answer to your question a bit here. Since I don't know of any unified treatment, I don't feel qualified to say "any time you see this phenomena, you should use an X-transformation." (other than what the books might say with regard to Fourier analysis on Groups and the fact that the transforms are (very) roughly equivalent...

I too am fascinated by the Laplace transform (and it's analogues). I'd recommend looking into Control Theory which is expands on the ideas of the Laplace Transform. It's usually treated in an "engineery" manner, but it is very much a mathematical theory.


EDIT: This is by no means the limits of transform methods, this only reflects my interests/knowledge, others will have many more examples.

EDIT2: I will again be teaching the Laplace this semester in DE, every time I do this, I wish I had more time to start an intro to Control Theory as it flows so naturally from the Laplace.