Shadows were available until FlightGear 1.0. The feature became deprecated with the OSG based FlightGear 1.9.0. However, the shift to OSG opens up the possibility for re-implementing shadows and many other new graphical features in future FlightGear versions (such as happened with 3D Clouds).
Implementing Shadow Mapping
Shadow mapping is a technique that use a texture where distance of objects from the light has been recorded. During rendering, the distance to the light of the fragment rendered is compared to the shortest distance of the scene to the light, and if it is higher, the fragment is drawn using only ambient light. Otherwise, it is fully lit.
Project Rembrandt is a new project that currently implements shadow mapping for FlightGear.
Another project which is in some parts related to shadow rendering is the Atmospheric Light Scattering (ALS) project.