I would rather turn the question round. Most "real world" structures turn out to be statically indeterminate because of the way they are constructed, whether you like it or not. For example you can design the roof structure of a house using statically determinate trusses, but the walls of a traditional (european) house built from brick or stone are not statically determinate. Even with a modern steel-framed building, including the diagonal members that are needed to make a rigid statically determinate frame would be a problem, unless you like triangular windows, or beams running diagonally across the rooms, etc!
On the other hand statically determinate structures have one big advantage. The internal forces and reaction forces only depend on the geometry of the structure, and not what it is made from. So it is easy to design a "minimum weight" determinate structure: calculate the internal forces, and then use the least amount of material that will withstand them. (OK, that ignores the weight of the structure itself, but you get the general idea).