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Atmospheric studies at high spectral resolution have shown the presence of molecules, neutral and ionised metals, and hydrogen in the transmission spectrum of ultra-hot Jupiters, and have started to probe the dynamics of their atmospheres. We analyse the transmission spectrum of MASCARA-1b, one of the densest ultra-hot Jupiters orbiting a bright (V=8.3) star. We focus on the CaII H&K, NaI, LiI, H$α$, and KI D1 spectral lines and on the cross-correlated FeI, FeII, CaI, YI, VI, VII, CaH, and TiO lines. For those species that are not present in the stellar spectrum, no detections are reported, but we measure upper limits with an excellent precision ($\sim10$ ppm for particular species). For those species that are present in the stellar spectrum and whose planet-occulted spectral lines induce spurious features in the planetary transmission spectrum, an accurate modelling of the Rossiter-McLaughlin effect (RM) and centre-to-limb variations (CLV) is necessary to recover possible atmospheric signals. In the case of MASCARA-1b, this is difficult due to the overlap between the radial velocities of the stellar surface regions occulted by MASCARA-1b and the orbital track along which the planet atmospheric signal is expected to be found. To try to disentangle a possible planetary signal, we compare our results with models of the RM and CLV effects, and estimate the uncertainties of our models depending on the different system parameters. Unfortunately, more precise measurements of the spin-orbit angle are necessary to better constrain the planet-occulted track and correct for the transit effects in the transmission spectrum with enough precision to be able to detect or discard possible planetary absorptions. Finally, we discuss the possibility that non-detections are related to the low absorption expected for a high surface gravity planet such as MASCARA-1b.