Inelastic collisions occur when two objects collide and stick together, resulting in a loss of kinetic energy. In the case of spherical rigid bodies, the conservation laws of linear momentum and kinetic energy can be applied to analyze the collision.

Linear Momentum:
Linear momentum is the product of an object's mass and its velocity. In an isolated system where no external forces act on the objects, the total linear momentum before and after the collision is conserved. In the case of inelastic collisions between two spherical rigid bodies, the linear momentum is conserved.

When the two bodies collide and stick together, the momentum is transferred between them. The total momentum before the collision is equal to the total momentum after the collision. This is because the combined mass of the two bodies remains the same, and their velocities change such that the total momentum is conserved.

Kinetic Energy:
Kinetic energy is the energy possessed by an object due to its motion. Unlike linear momentum, the conservation of kinetic energy is not applicable to inelastic collisions. In an inelastic collision, some of the kinetic energy is lost as internal energy, such as deformation or heat. The objects stick together after the collision, and their combined kinetic energy is lower than the initial total kinetic energy.

Since the question specifically mentions an inelastic collision, it is correct to state that the total mechanical energy is not conserved. Mechanical energy is the sum of kinetic energy and potential energy, and in an inelastic collision, there is a loss of kinetic energy.

Conclusion:
In summary, when considering an inelastic collision between two spherical rigid bodies:
- The linear momentum is conserved because the total momentum before and after the collision remains the same.
- The kinetic energy is not conserved because some of it is lost as internal energy.
Therefore, the correct answer to the given question is option 'D' - the linear momentum is conserved.