How does the shape (form; not cross-section) of a spring impact performance? Cylindrical compression springs are everywhere and then some applications choose other forms like rectangular or unique polygonal form. What impact does the form of a compression spring have and how do you calculate an optimal solution?
 A: I could find a site which describes about the variations in spring designs, mainly attributed to the usage of the spring
.
The resource describes the conical spring, double conical springs from the normal cylindrical springs.. The site says and I quote,

Conical springs provide a commonly used solution for spring
  applications with constraints of reduced length or space. They can be
  used in many different mechanisms, such as contactors and switches in
  the electrical field. Indeed, they are often chosen for one special
  characteristic: their ability to telescope. They take up very little
  space at maximum compression while storing as much energy as
  cylindrical springs. Their load-length characteristics are usu­ally
  nonlinear.

A: When you compress a spring you are twisting the rod that makes up the spring. A cylindrical spring twists the whole coil evenly, so you get a linear response and minimal deformation of the rod that makes up the spring. This means you can make the spring as light (and cheap) as possible.
Any other design will twist the rod that makes up the spring unevenly, so you need a greater weight of metal for the same spring strength. As Vineet mentioned, using different designs can give advantages such as allowing greater compression. Weird designs such as rectangular springs are (I'm guessing really) dictated by overall design requirements e.g. making the spring fit in a restricted space.
A: The engineering point of view assumably gives a lot of freedom in the spring design. The www.industrial-springs.com gives examples of different type of springs with the different parameters:
Compression springs:
The springs are intended for tests and experiments. You can cut them to the required length. The ends of the compression springs should be closed after the desired length is found.
Conical Compression springs:
This series of conical compression springs is designed to provide travel all the way down to a height (Ln) of 2 x wire thickness. The spring constant is linear due to the variable pitch.
Multiwave Compression springs:
Multiwave compression springs are designed to be used in applications where space problems in the length preclude the option of using a "normal" compression spring.
