Charge is distributed uniformly over the surface of a uniform conductor. If the surface is flat or spherical the the charge is uniformly distributed. Most surfaces do not have this kind of symmetry. It is more general to say that charge distributes itself so that the potential is uniform. This means that charge tends to concentrate in areas of maximum curvature, or corners.

This means that the electric field is more intense in areas of maximum curvature.

This then implies that the electric field is greatest at those points on the surface where the charge distribution is greatest.

If the electric field breaks down and discharge occurs, it will be at those points of maximum electric field ie greatest curvature.