Aims. Abundances of the Mg isotopes 24 Mg, 25 Mg, and 26 Mg can be used to test models of chemical enrichment of interstellar/intergalactic gas clouds. Additionally, because the position of the Mg ii λλ2796, 2803 Å lines is often taken as a reference in computations of possible changes of the fine-structure constant α, it should be clarified to which extent these lines are affected by isotopic shifts. Methods. We use a high-resolution spectrum (pixel size ≈1.3 km s −1 ) of the quasar HE0001-2340 observed with the UVES/VLT to measure Mg isotope abundances in the intervening absorption-line systems at high redshifts. Line profiles are prepared taking into account possible shifts between the individual exposures. In the line-fitting procedure, the lines of each ion are treated independently. Because of the unique composition of the selected systems -the presence of several transitions of the same ion -we can test the local accuracy of the wavelength scale calibration, which is the main source of errors in the sub-pixel line position measurements. Results. In the system at z abs = 0.45, which is probably a fragment of the outflow caused by SN Ia explosion of highmetallicity white dwarf(s), we measured velocity shifts of Mg ii and In the systems at z abs = 1.58 and z abs = 1.65 enriched by AGB-stars we find only upper limits on the content of heavy Mg isotopes ( 25 Mg+ 26 Mg)/ 24 Mg < ∼ 0.7 and ( 25 Mg+ 26 Mg)/ 24 Mg < ∼ 2.6, respectively. At z abs = 1.58, we also put a strong constraint on a putative variation of α: Δα/α = (−1.5 ± 2.6) × 10 −6 , which is one of the most stringent limits obtained from optical spectra of QSOs. We reveal that the wavelength calibration in the range above 7500 Å is subject to systematic wavelength-dependent drifts.