This work focuses on the study of the anomalous microwave emission (AME), an important emission mechanism between 10 and 60 GHz whose polarisation properties are not yet fully understood and is therefore a potential contaminant for future cosmic microwave background (CMB) polarisation observations. We used new QUIJOTE-MFI maps at 11, 13, 17, and 19 GHz obtained from the combination of the public wide survey data and additional 1800 h of dedicated raster scan observations together with other public ancillary data, including WMAP and Planck, to study the polarisation properties of the AME in three Galactic regions: ρ Ophiuchi, Perseus, and W43. We obtained the spectral energy distributions (SEDs) of the three regions over the frequency range 0.4–3000 GHz in intensity and polarisation. The intensity SEDs are well described by a combination of free-free emission, thermal dust, AME, and CMB anisotropies. In polarisation, we extracted the flux densities using all available data between 11 and 353 GHz. We implemented an improved intensity-to-polarisation leakage correction that allowed reliable polarisation constraints well below the 1% level from Planck-LFI data to be derived for the first time. A frequency stacking of maps in the range 10–60 GHz allowed us to reduce the statistical noise and to push the upper limits on the AME polarisation level. We obtained upper limits on the AME polarisation fraction of the order ≲1% (95% confidence level) for the three regions. In particular, we obtained ΠAME < 1.0% (at 28.4 GHz), ΠAME < 0.9% (at 28.4 GHz), and ΠAME < 0.28% (at 33 GHz) in ρ Ophiuchi, Perseus, and W43, respectively. At the QUIJOTE 17 GHz frequency band, we found ΠAME < 5.0% for ρ Ophiuchi, ΠAME < 3.4% for Perseus, and ΠAME < 0.85% for W43. We note that for the ρ Ophiuchi molecular cloud, the new QUIJOTE-MFI data allowed us to set the first constraints on the AME polarisation in the range 10–20 GHz. Our final upper limits derived using the stacking procedure are ΠAME < 0.58% for ρ Ophiuchi, ΠAME < 0.67% for Perseus, and ΠAME < 0.31% for W43. Altogether, these are the most stringent constraints to date on the AME polarisation fraction of these three star-forming regions.