Detecting planets in the early stages of formation is key to reconstructing the history and diversity of fully developed planetary systems. The aim of this work is to identify potential signatures from planet-disc interaction in the circumstellar discs around MWC 480, HD 163296, AS 209, IM Lup, and GM Aur, through the study of molecular lines observed as part of the ALMA large program MAPS. Extended and localised perturbations in velocity, line width, and intensity have been analysed jointly using the DISCMINER modelling framework, in three bright CO isotopologues, 12CO, 13CO, and C18O J = 2−1, to provide a comprehensive summary of the kinematic and column density substructures that planets might be actively sculpting in these discs. We find convincing evidence for the presence of four giant planets located at wide orbits in three of the discs in the sample: two around HD 163296, one in MWC 480, and one in AS 209. One of the planet candidates in HD 163296, P94, previously associated with velocity signatures detected in lower velocity resolution 12CO data, is confirmed and linked to localised velocity and line width perturbations in 13CO and C18O too. We highlight that line widths are also powerful tracers of planet-forming sites as they are sensitive to turbulent motions triggered by planet-disc interactions. In MWC 480, we identified non-axisymmetric line width enhancements around the radial separation of candidate planet-driven buoyancy spirals, which we used to narrow the location of the possible planet to an orbital radius of R = 245 au and PA = 193°, at a projected distance of 1.33″ from the star. In the disc of AS 209, we found excess 12CO line widths centred at R = 210 au, PA = 151°, at a projected distance of 1.44″, spanning around the immediate vicinity of a circumplanetary disc candidate proposed previously, which further supports its presence. We report no clear localised or extended kinematic signatures in the discs of IM Lup or GM Aur that could be associated with the presence of planets or gravitational instabilities. On the other hand, we demonstrate that pressure minima exhibit line width minima counterparts in optically thick emission, making them robust tracers of gaps in the gas surface density when analysed together with azimuthal velocity flows. Finally, we show that nine out of eleven millimetre dust continuum rings in the sample are co-located with pressure bumps traced by kinematic modulations, indicating that aerodynamic confinement via pressure traps is a common mechanism for the formation of dust substructures in these discs. Overall, our analysis reveals that all discs in the sample present a remarkable level of substructure in all the line profile observables considered, regardless of the CO isotopologue. However, the magnitude and morphology of the substructures vary between discs and tracers, indicating that the kinematics and thermodynamic properties are likely shaped by different physical mechanisms in each object. We propose that the main kinematic signatures identified in the discs of MWC 480, HD 163296, and AS 209 have a planetary origin, although they do not always manifest as highly localised perturbations, while the discs of IM Lup and GM Aur do not yield clear signatures pointing to the presence of massive planets. Our simultaneous analysis of multiple tracers and observables aims to lay the groundwork for robust studies of molecular line properties focused on the search for young planets in discs.