Which of the following is not true for a perfect conductor?

For a perfect conductor in electrostatic equilibrium, the electric field inside the conductor is zero, which is a fundamental property. This implies that the potential is constant throughout the volume of the conductor, making the surface an equipotential surface. Consequently, the electric field just outside the conductor must be perpendicular to the surface to ensure no tangential component exists, which would otherwise cause charge movement.

Regarding the distribution of charge, it resides entirely on the surface of the conductor. However, the charge density is not necessarily uniform. For a conductor of arbitrary shape, the surface charge density $\sigma$ is higher at regions with smaller radii of curvature, such as sharp points or edges, to maintain a constant potential. Therefore, the statement that charge is always uniformly distributed irrespective of the shape is false.

Answer: The charge carried by a conductor is always uniformly distributed over the surface of the conductor irrespective of the shape of the conductor.