If the magnetic field doesn't polarize does it follow the electric field path of propagation? or does it vanish?
The magnetic field polarizes orthogonal to the electric field in free space. We generally only talk about the electric field because Maxwell's equations define a one to one relationship between the two. It would make just as much sense to only talk about the magnetic field. We choose the electric field because, in general, when light interacts with matter it is the electric field which causes all of the interesting effects (though this in not strictly true).
The magnetic field does not vanish when light is polarized. A changing electric field induces a magnetic field, and a changing magnetic field induces an electric field. This is why, in the propagation of an electromagnetic wave, there is always an oscillating electric field coupled with a magnetic field oscillating perpendicular to this electric field. You cannot simply take away one of these fields.
Unpolarized light consists of many electromagnetic waves polarized in different directions. Each of these waves has its own electric and magnetic fields, which are perpendicular to each other. When this light is polarized by, say, sending it through a polarizing filter, the electromagnetic wave that results is polarized only in one direction. There still exists both an electric and magnetic field to this wave.
The graphic below illustrates this effect, but only shows the electric field. In reality, there is still a magnetic field that oscillates perpendicularly to the resulting, polarized electric field.