We present an experimental investigation of slow transients in the gate and drain currents of MoS2-based transistors. We focus on the measurement of both the gate and drain currents and, from the comparative analysis of the current transients, we conclude that there are at least two independent trapping mechanisms: trapping of charges in the silicon oxide substrate, occurring with time constants of the order of tens of seconds and involving charge motion orthogonal to the MoS2 sheet, and trapping at the channel surface, which occurs with much longer time constants, in particular when the device is in a vacuum. We observe that the presence of such slow phenomena makes it very difficult to perform reliable low-frequency noise measurements, requiring a stable and repeatable steady-state bias point condition, and may explain the sometimes contradictory results that can be found in the literature about the dependence of the flicker noise power spectral density on gate bias.