Aims. Recently, several arguments have been presented that favour a scenario in which the low detection rate of magnetic fields in Herbig Ae stars can be explained by the weakness of these fields and rather large measurement uncertainties. Spectropolarimetric studies involving sharp-lined Herbig Ae stars appear to be a promising approach for the detection of such weak magnetic fields. These studies offer a clear spectrum interpretation with respect to the effects of blending, local velocity fields, and chemical abundances, and allow us to identify a proper sample of spectral lines appropriate for magnetic field determination. Methods. High-resolution spectropolarimetric observations of the three sharp-lined (vsini< 15 km s^-1) Herbig Ae stars HD 101412, HD 104237, and HD 190073 have been obtained in recent years with the HARPS spectrograph in polarimetric mode. We used these archival observations to investigate the behaviour of their longitudinal magnetic fields. To carry out the magnetic field measurements, we used the multi-line singular value decomposition (SVD) method for Stokes profile reconstruction. Results. We carried out a high-resolution spectropolarimetric analysis of the Herbig Ae star HD 101412 for the first time. We discovered that different line lists yield differences in both the shape of the Stokes V signatures and their field strengths. They could be interpreted in the context of the impact of the circumstellar matter and elemental abundance inhomogeneities on the measurements of the magnetic field. On the other hand, due to the small size of the Zeeman features on the first three epochs and the lack of near-IR observations, circumstellar and photospheric contributions cannot be estimated unambiguously. In the SVD Stokes V spectrum of the SB2 system HD 104237, we detect that the secondary component, which is a T Tauri star, possesses a rather strong magnetic field â B_zâ