Find a copy of the Morse & Feshbach book on Theoretical Physics (volume 1). They gave nice stereographs of the different coordinate systems (and encourage you to learn how to be able to look at a stereograph without using an optical viewer, as it's easy once learned). I haven't seen that book for 40 years, but they may have given examples such as separating Laplace's equation or somesuch in each of the different coordinate systems.
Edit: check Papabravo's link to wikipedia, as it outlines what I suggested...