Quandaries and Queries
Who is asking: Student
There are so many possible answers here.
Spherical geometry is applicable to all kinds of navigation and related calculations for movement on the earth (at least as a first approximation).
A more unusual application, which is still a subject of geometry research, is in computational origami. Consider a piece of paper folded with all folds through a single point. Can the paper be flattened without using any other folds except those final folds, without cutting the paper? Conversely, given a 'design' with the correct angle of paper around the vertex (that is 360 degrees) can the design be folded without additional folds? The proof that the answer is yes (being written up right now) using spherical geometry - since you might as well imagine the vertex is the center of the sphere, and you have cut off the paper with a circle centered at this vertex. The edges of the paper are now line segments on the sphere, and you are looking at motions of the polygon of such edges, on the sphere. If this seems esoteric, a young Canadian Geometer, Erik Demain (an assistant professor at MIT) just won a McArthur 'Genius Award' of half a million work, in part for his related work in computational origami, and related problems in the plane.
Hyperbolic geometry is less obvious - but very important. It turns out if you want to classify all the possible patterns of symmetries and 'tiling's which can occur, many of them are not on the plane, or on the sphere, but most occur in the hyperbolic plane. These can then be mapped onto various surfaces which occur in 3-space. If this sounds abstract, a recent workshop on designing new materials had several presentations about how to use this type of thinking to understand possible real and artifical crystal structures. They are looking for new materials which will store more hydrogen (.e.g for cars using fuel cells) or will absorb more toxic metals (such as radioactive material released in accidents at reactors). I was amazed - but these were chemists and material science people talking.
More generally, hyperbolic geometry is closely related to the geometry of space as it is described in the physics of general relativity. Something invented to respond to an intellectual puzzle turns out to be what we need to understand real physical problems!