In regular three dimensional space we always limit ourselves to Cartesian (i. e. orthonormal) frames. This has lots of advantages: dot products are easy, no need to distinguish between vectors and covectors, finding components of vectors is simple, etc. Even when using curvilinear coordinates, our basis vectors are orthonormal.
Of course linear algebra tells us that we can use whatever basis we want. So I ask: is there any situation in things are easier in a non Cartesian frame?
Edit: I'm not talking about coordinates. As mentioned in the comments, lots of different coordinate systems (spherical, cylindrical, etc) are used when there is a special geometry that makes things easier. But at least in spherical and cylindrical coordinates, the basis vectors at each point are orthonormal. I want to know whether there are reasons for using basis vectors that are not orthonormal.