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Source models of mid-oceanic earthquakes are often based only on far-field, teleseismic data. The uncertainties of all source parameters are rarely quantified, which restricts our understanding of how these events slip and how oceanic lithosphere is formed. Here, we perform moment tensor inversions for five M w 4.6–5.9 earthquakes that occurred in the Azores archipelago near the Mid-Atlantic Ridge in 2013–2016, taking advantage of the recently expanded seismic network in the region. We assess moment tensor uncertainties due to data and Earth model variability as well as the robustness of teleseismic versus local data inversions. We find that for the events studied: (i) existing 1-D Earth models of the region based on receiver function data lead to a slightly improved data fit of local data compared to a widely used regional model based on active seismic surveys; and (ii) using different 1-D Earth models in the local data inversions leads to a variability in the retrieved source parameters of 15°–30° in fault strike, 5°–20° in dip, and 20°–60° in rake, depending on the earthquake's magnitude and location. We study in detail the M w 5.9 2013 April 30 Povoação basin earthquake using 1-D and 3-D waveform modelling, for which reported values of strike, dip, and rake in earthquake catalogues differ by 60°, 35°, and 80°. We find that our moment tensor solutions show a lower variability than in the catalogues and exhibit a persistent non-double-couple component of ∼40–60  per cent, which is not due to a volumetric change. We suggest that it is potentially due to geometrically complex faulting in the Povoação basin, notably curved faults. We find that the retrieved moment tensor solutions depend strongly on the earthquake's location. If an accurate location is used, joint inversions of local and teleseismic data can help to stabilize moment tensor solutions of oceanic earthquakes and reduce parameter trade-offs, compared to inversions of local data alone. [ABSTRACT FROM AUTHOR]