We report a study of the basins of attraction for potential energy minima defined by different minimization algorithms for an atomic system. We find that whereas some minimization algorithms produce compact basins, others produce basins with complex boundaries or basins consisting of disconnected parts. Such basins deviate from the "correct" basin of attraction defined by steepest-descent pathways, and the differences can be controlled to some extent by adjustment of the maximum step size. The choice of the most convenient minimization algorithm depends on the problem in hand. We show that while L-BFGS is the fastest minimizer, the FIRE algorithm is also quite fast and can lead to less fragmented basins of attraction.