The ground plane theory doesn't matter whether we're talking FM or AM modulation, but the frequencies (actually, the wavelengths) involved are what matter.

Without boring people to tears with a load of radio theory, here's the general rule of when ground planes start to matter:

When you have an "unbalanced" vertical antenna (say, a 1/4 wavelength or 5/8ths wavelength whip), these simple whip vertical antennas really need a ground plane to "anchor" the other side of the RF energy being sent into the whip. ("anchor" isn't the technical radio term, but it is the best non-radio-geek term I can come up with).

The idea is that a vertical works best when you have the outside of the coax grounded, preferably to a real ground, with a conductive mesh that radiates outwards infinitely in all directions equally around the vertical radiator.

Well, that's theory, here's what will get you by: If you have a conductive surface that is at least 1/4 wavelength in radius around the centerpoint of the vertical antenna, you get the vast of the benefit of a perfect ground. If we're talking of the 158 to 172 MHz business band, that would be 16" in radius or so. If you can create a disk of 1/2 wavelength in radius, you get an even better ground plane.

Now, if you have a non-symmetric ground plane, you will get better radiation from the vertical in the direction of the largest area of ground plane conductor.

The conductor for the ground plane doesn't have to be solid -- it could be a mesh, just as long as the holes in the mesh are sufficiently small (say, less than 1/16th of a wavelength). Solid does work better, but we're talking marginal improvements here.

Here's the kicker: everything I've just said about ground planes applies to transmitting antennas. In receiving radio antennas, a vertical antenna gets a little bit better reception with a ground plane, but it isn't a huge difference. 

More than likely, the reason why your FM radio isn't receiving well is that the antenna trimmer on the receiver needs to be adjusted to match your antenna, or you have coax that has gotten moisture into the cable.