The SHARAD (shallow radar) sounding radar on the Mars Reconnaissance Orbiter detects subsurface reflections in the eastern and western parts of the Medusae Fossae Formation (MFF). The radar waves penetrate up to 580 m of the MFF and detect clear subsurface interfaces in two locations: west MFF between 150 and 155° E and east MFF between 209 and 213° E. Analysis of SHARAD radargrams suggests that the real part of the permittivity is ~3.0, which falls within the range of permittivity values inferred from MARSIS data for thicker parts of the MFF. The SHARAD data cannot uniquely determine the composition of the MFF material, but the low permittivity implies that the upper few hundred meters of the MFF material has a high porosity. One possibility is that the MFF is comprised of low-density welded or interlocked pyroclastic deposits that are capable of sustaining the steep-sided yardangs and ridges seen in imagery. The SHARAD surface echo power across the MFF is low relative to typical martian plains, and completely disappears in parts of the east MFF that correspond to the radar-dark Stealth region. These areas are extremely rough at centimeter to meter scales, and the lack of echo power is most likely due to a combination of surface roughness and a low near-surface permittivity that reduces the echo strength from any locally flat regions. There is also no radar evidence for internal layering in any of the SHARAD data for the MFF, despite the fact that tens-of-meters scale layering is apparent in infrared and visible wavelength images of nearby areas. These interfaces may not be detected in SHARAD data if their permittivity contrasts are low, or if the layers are discontinuous. The lack of closely spaced internal radar reflectors suggests that the MFF is not an equatorial analog to the current martian polar deposits, which show clear evidence of multiple internal layers in SHARAD data.