We present a method which uses cuts in colour-colour and reduced proper motion-colour space to select white dwarfs without the recourse to spectroscopy while allowing an adjustable compromise between completeness and efficiency. Rather than just producing a list of white dwarf candidates, our method calculates a probability of being a white dwarf (P_WD_) for any object with available multiband photometry and proper motion. We applied this method to all objects in the Sloan Digital Sky Survey (SDSS) Data Release 10 (DR10) photometric footprint and to a few selected sources in DR7 which did not have reliable photometry in DR9 or DR10. This application results in a sample of 61 969 DR10 and 3799 DR7 photometric sources with calculated P_WD_ from which it is possible to select a sample of ~23000 high-fidelity white dwarf candidates with T_eff_>~7000K and g<=19. This sample contains over 14000 high confidence white dwarfs candidates which have not yet received spectroscopic follow-up. These numbers show that, to date, the spectroscopic coverage of white dwarfs in the SDSS photometric footprint is, on average, only ~40 percent complete. While we describe here in detail the application of our selection to the SDSS catalogue, the same method could easily be applied to other multicolour, large area surveys. We also publish a list of 8701 bright (g<=19) white dwarfs with SDSS spectroscopy, of which 1781 are new identifications in DR9/DR10.