  A study of the Fermi surface of the putative topological semimetal Pd$_3$Pb
has been carried out using Shubnikov-de Haas (SdH) oscillations measured in
fields of up to 60 T. Pd$_3$Pb has garnered attention in the community due to a
peculiar Fermi surface that has been proposed theoretically by Ahn, Pickett,
and Lee, [Phys. Rev. B 98, 035130 (2018)] to host a dispersion-less band along
$X-\Gamma$ as well as multiple triply-degenerate band crossings that, under the
influence of spin-orbit coupling, lead to ten four-fold degenerate Dirac
points. Analysis of the SdH oscillation data verifies the calculated
multi-sheet Fermi surface, revealing a $\Gamma$ centered spheroid that had not
been resolved experimentally in prior studies. A comprehensive, angle-dependent
analysis of the phase of the SdH oscillations convincingly demonstrates a
non-trivial Berry phase for two bands along $\Gamma-R$, supporting the
theoretical predictions, while simultaneously evidencing interference between
extremal orbits that mimics a trivial Berry phase at intermediate angles.
