Since the rapid technological progress demands for ever smaller storage units, the emergence of stable magnetic order in nanomaterials down to the single-atom regime has attracted huge scientific attention to date. Electronic transport spectroscopy has been proven to be a versatile tool for the investigation of electronic, magnetic, and mechanical properties of atomic contacts. Here we report a comprehensive experimental study of the magnetoconductance and electronic properties of Pd atomic contacts at low temperature. The analysis of electronic transport (d/ fdV) spectra and the magnetoconductance curves yields a diverse behavior of Pd single-atom contacts, which is attributed to different contact configurations. The magnetoconductance shows a nonmonotonous but mostly continuous behavior, comparable to those found in atomic contacts of band ferromagnets. In the dl fdV spectra, frequently, a pronounced zero-bias anomaly (ZBA) as well as an aperiodic and nonsymmetric fluctuation pattern are observed. While the ZBA can be interpreted as a sign of the Kondo effect, suggesting the presence of magnetic impurity, the fluctuations are evaluated in the framework of conductance fluctuations in relation to the magnetoconductance traces and to previous findings in Au atomic contacts. This thorough analysis reveals that the magne~oconductance and transport spectrum of Au atomic contacts can completely be accounted for by conductance fluctuations, while in Pd contacts the presence of local magnetic order is required.