A spring can have many geometrical shapes. A spring is a component that deforms elastically under a mechanical load and where the relation between elastic deformation and load is important.
The relation between load and deflection is called the spring characteristic or the spring rate. When mechanical systems are developed and a need for a spring function arises, we define the requirement on the relation between load and deformation.
The type of spring and its geometry is chosen based on the geometrical space that is available for the spring function, together with considering reliability and cost. There are some spring types that have proven to be cost and space effective ways of realizing certain spring functions. Helical coil springs, disc springs and wave springs are normally used for axial compression loads and each of them has their own range of force-deflection relation where they give the most space effective solution.
It is similar for rotating deflection, where helical torsion springs, clock springs, torsion bars or power springs each has their own optimum range of torque-deflection relation. But a component that has a built-in spring function can have practically any shape. Wire and sheet can be formed to complicated shapes and many such components have more than one function, of which only one is a spring function.