Padé approximants are employed in order to study the analytic structure of the four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, are used as
fitting functions to lattice data for the propagators, carefully propagating uncertainties due to the fit procedure and taking into account all possible correlations. Applying this procedure systematically to the gluon-propagator data, we observe the presence of a pair of complex poles at $p^2_{\mathrm{pole}} = (-0.37 \pm 0.05_{\mathrm{stat}}
\pm 0.08_{\mathrm{sys}}) \pm \, i\, (0.66 \pm 0.03_{\mathrm{stat}}\pm 0.02_{\mathrm{sys}}) \, \mathrm{GeV}^2$, where ''stat'' represents the statistical error and ''sys'' the systematic one. We also find a zero on the negative real axis of $p^2$, at $p^2_{\mathrm{zero}} =
(-2.9 \pm 0.4_{\mathrm{stat}} \pm 0.9_{\mathrm{sys}}) \, \mathrm{GeV}^2$. We thus note that our procedure --- which is based on a model-independent approach and includes careful error propagation --- confirms the presence of a pair of complex poles in the gluon propagator, in agreement with previous works. For the ghost propagator, the Padés indicate the existence of the single pole at $p^2 = 0$, as expected. We also find evidence of a branch cut on the negative real axis. Through the use of the so-called D-Log Padé method, which is designed to approximate functions with cuts, we corroborate the existence of this cut for the ghost propagator.