Nowadays one has to have a sneaky reason to find out.

No, one doesn't. Anybody can find out the tax and MoT status of any vehicle here:

Enter the registration number of the vehicle - Check if a vehicle is taxed and has an MOT - GOV.UK (vehicleenquiry.service.gov.uk)

Sorry, I get where you're coming from.

Possibly the OP is a bit miffed that this driver is freeloading and taking up parking space in his road. I would be too.

The DVLA will eventually get round to this. They don't have to witness the car being used. All vehicles must be either either taxed (and insured) or declared (and kept) off road by way of a SORN. The Registered Keepers of any that are not are committing an offence which the DVLA will action either by an out-of-court settlement or prosecution (eventually).

There was a Toyota MR2 left in our road for weeks without being moved. The brake discs wew so rusty they looked like wood. Upon checking on line it was subject to SORN. Neighbour was fairly sue it belonged to a house on the next road with a long garden and a gate at rear. Apparently there was a 'menagerie of old cars' on their land.

I think he reported it to DVLA who duly visited and clamped it. Cost him several hundred pounds in penalties etc. Since then it's kept taxed.

I think it's a question of them being aware of where the car is at a time when they're agents are working.

Is the OP aware of whether the car he complains of is insured?

There's a car on the road near me with a council 'abandoned car' sticker on the window. It's been like that for weeks. It's taxed, but not insured or MOT'd. I suppose 'day insurance' could've been taken out while it was still MOT'd, enabling it to be taxed. I expect it'll stay there until the cows come home. DVLA aren't going to be interested, because it's taxed. BIBs aren't interested unless it's being driven without insurance. Nobody's interested about the expired MOT. I suppose the council don't want to go to the expense of removing it without re-couping their costs and are concerned it may leave them open to being sued if they do!

Round here Council and Police are both pretty good at getting abandoned and uninsured cars off the road. When I worked in the office in the centre of Northampton it was fairly common to see cars being picked up by contractors for the Police because they'd pinged ANPR while stopped at the side of the road.

West Northants Council regularly have stuff on facebook about cars lifted from streets - mostly over on the east sdie of town.