Knowing the geodesic structure of a given spacetime has many observational impact. For instance, it allows to computed the redshift and deformation of light emitted close to the central object.
This is how the spin parameter of supermassive black hole is estimated. The accretion disk of supermassive BH is mostly composed of an iron plasma that emits a X-ray signal thought its Kalpha line. Due to the bending of spacetime close to the black hole, this X-ray line is redshifted, leading to a distorted X-ray spectrum. The distortion is a function of many parameters, including the spin parameter of the black hole, the shape of the accretion disk…The first estimate (for SgA*) led to almost extremal BH. Note that it has been recently argued that assuming a non Kerr BH (e.g. in alternative theory of gravity) lead to similar effect in the spectrum distortion than almost extremal BH. See the nice video from Nasa that illustrate this.
Another interesting application is related to Quasi-periodic oscillations. While they are not very well understood for NS, their origin for black holes is expected to be related to oscillation modes of the geodesics.
The survey of motion around the active galactic nucleus in the center of the galaxy led to estimating the nature of the central object. Sagitarius A* is now widely believed to be a black hole.
Just for fun, this is a video of a 10 body simulation with the parameters of the solar system : http://youtu.be/Lfp-WFFdiuA
Out of this simulation, the perihelion advance of Mercury can be computed and agrees very well with the historical Newtonian prediction of 531.7 ” / 100 years: