We investigate the analytic structure of the one-loop boson propagator at finite density and temperature in hadronic and QCD plasmas. The appearance of poles in the complex wave-vector plane dramatically modifies the form of the screened static potential of 'charges' in the medium, which substantially differs from the standard Debye form. In fact, at T = 0, within a wide range of distance, the potential for two interacting particles is dominated by the pole contribution, which leads to an oscillatory and exponentially damped form (Yukawa oscillations). For the long range, the form of the potential is dominated by the Kohn singularity, which induces oscillatory behavior and is damped as a power-law of the distance (Friedel oscillations). We discuss the evolution of the pole, as the density and temperature change, for the two types of plasmas we have analyzed. In both cases, these phenomena have important consequences when analyzing the in-medium interaction.