Wide-field photometric surveys enable searches of rare yet interesting objects, such as strongly lensed quasars or quasars with a bright host galaxy. Past searches for lensed quasars based on their optical and near-infrared properties have relied on photometric cuts and spectroscopic preselection (as in the Sloan Quasar Lens Search), or neural networks applied to photometric samples. These methods rely on cuts in morphology and colours, with the risk of losing many interesting objects due to scatter in their population properties, restrictive training sets, systematic uncertainties in catalogue-based magnitudes and survey-to-survey photometric variations. Here, we explore the performance of a Gaussian mixture model to separate point-like quasars, quasars with an extended host and strongly lensed quasars using grizpsf and model magnitudes and WISEW1, W2. The choice of optical magnitudes is due to their presence in all current and upcoming releases of wide-field surveys, whereas UV information is not always available. We then assess the contamination from blue galaxies and the role of additional features such as W3 magnitudes or psf-model terms as morphological information. As a demonstration, we conduct a search in a random 10 per cent of the SDSS footprint, and provide the catalogue of the 43 SDSS object with the highest 'lens' score in our selection that survive visual inspection, and are spectroscopically confirmed to host active nuclei. We inspect archival data and find images of 5/43 objects in the Hubble Legacy Archive, including two known lenses.