Disc and Cylinder magnets can be axially or diametrically magnetized.
Block magnets are defined by three dimensions: Length, Width and Thickness. To be consistent, we define the Thickness dimension along the axis of magnetization. The thickness is usually the smallest dimension, but not always!
We're sometimes asked if we can provide block magnets magnetized through the length or width. Yes, we do have some long "bar" magnets, where the magnetization is in the longest dimension. We do still define this as the Thickness, though.
Ring magnets can be axially or diametrically magnetized.
Spheres can only be axially magnetized. There is only one possible direction.
Arc Segment magnets can be magnetized in one of four directions.
One example is the AY0X030-C.
One example is the AY0X030-T.
Note that the magnetization is along a straight axis, and is not truly radial. The lines do not converge at the center of the arc. When assembled, arc segments like this can be used to approximate a radially magnetized ring.
Again, the lines of magnetization are all parallel. The lines do not curve around the arc.