Does an inverted-L benefit from having radials running out 360 degrees from the base of the vertical section much as a vertical 1/4 wave antenna does?
All forms of vertical, end-fed radiators such as used by stations in the AM broadcast band and by some amateur operators need a good "r-f ground" in order to radiate as efficiently as possible.
The transmitter provides a source for the r-f current, and the r-f ground provides the completion of the path caused by radiation from the monopole needed for those currents to return to the ground side of the transmitter circuits.
A monopole antenna system can be thought of as a series circuit where energy circulates back and forth at an r-f rate between the radiator and the r-f ground connection. The path between a monopole and an r-f ground is produced by the capacitance of the monopole to the r-f ground, across which path displacement currents flow.
Those displacement currents become conducted currents at, and just below the surface of the earth out to about 1/2-wavelength from a monopole, regardless of the monopole height in wavelengths. Once those currents enter the earth they need to be conducted back to the ground terminal of the transmitter in order to complete the path needed for r-f current to flow in the antenna system.
Soil is a rather poor (lossy) conductor of radio waves, so a system of buried radial wires often is used to provide a low-resistance path back to the transmit system in the area of the earth where those currents are highest -- within 1/2-wavelength radius of the monopole.
Carefully done physical experiments back in the 1930s determined that an r-f ground consisting of about 120 buried radials spaced 3 degrees apart around the tower base, and each radial about 0.4 wavelengths long (in free space) would produce an antenna system that radiated about 95% of the power applied to it for monopole heights of about 45 degrees or more. These results are produced regardless of the actual earth conductivity at, and within 1/2-wavelength of the base of monopole.
Such an r-f ground is the norm for licensed AM broadcast stations, but probably would be judged as too expensive, or even unnecessary by most amateur operators.
It should be noted that an r-f ground consisting of one or more "ground rods" buried vertically at or near the base of a monopole, or a few buried radial wires of any length do not constitute a good r-f ground. The r-f resistance of such paths is very high to the r-f earth currents surrounding the monopole, because they are forced to travel long paths through the lossy earth from up to 1/2-wavelength away to reach those conductors.
For reference, the r-f resistance of a broadcast type buried radial system is 2 ohms or less, while the r-f resistance of a few buried ground rods or wires may be 50 ohms or more. This added loss makes a big difference in the percentage of available r-f energy that will be radiated by a monopole, especially if the monopole is electrically short (ie, has low radiation resistance).
Note that a functional r-f ground as discussed here does not exist along the vertical length in space of any wire or other conductor such as a tower, billboard, water tower etc that is connected to a true r-f ground buried in the earth. All of those conductors will radiate into free space as a result of the r-f current flowing along them. And by definition, an r-f ground does not and cannot radiate.
Instead those exposed, vertical "ground" conductors become a radiating part of the antenna system. In some cases they can radiate more than the vertical conductor considered to be "the antenna."
A NEC-2D evaluation of this has been done for Part 15 MW AM installations (link below), and the same principles apply to HF systems in the amateur radio service.http://i62.photobucket.com/albums/h85/rfry-100/AM_System_Comparison.gif
R. Fry http://rfry.org