Registrations containing the letter 'Q' are very rare.
The letter 'Q' has been used since 1983 as the Prefix letter for vehicles of indeterminable age, such as Kit Cars, or other vehicles with substantial modifications, and/or parts from different donor vehicles, Stolen/Recovered Cars and some imports, etc, where the date of original manufacture can not be established. (see here: other numbers; Q-Regs)
Although not as common a site as they were in the late eighties and early nineties, Q plates have been making a bit of a comeback since 2011, when many mobility scooters have been given a vehicle registration for taxation purposes. Although these mobility scooters are not required to display a registration plate many owners do display them.
A scheme was also introduced for owners of off-road vehicles, mainly motorcycles, quadricycles, and construction/plant machinery where the owners could register their vehicle with the DVLA for added security and to improve detection rates if they do get stolen. The scheme was launched as a voluntary one, and the vehicles in question receive a Q registration.
The letter 'Q' has been also been used, within specially issued plates such as 'Diplomatic issues', 'temporary import series'.
Since 1997, when the Qatari royal family (and other wealthy Qataris) bring their cars to the UK and Europe, they use British-style plates with the letters 'Q', 'QR' or 'QTR' - the style and layout varies quite considerably.
'Q-Regs' are not all that common a site, and so we will display all 'Q' Regs!