I'm not really up on current models of mobility scooters, but as a general rule unless it's going to only be used on relatively flat surfaces (IE shopping centres etc), I'd always suggest going for a 4 wheeler, my mother went through multiple 3 wheelers and several neighbours had 3 wheelers and none of them seemed to be particularly stable in real world use, if say you were going along an average pavement (IE potentially uneven, with the little dips and rises you get), they were fine in say a shopping centre but not good for general use.
Also go with inflatable tyres, not solid where possible, solid tyres tend to make for a very uncomfortable run but are more durable* so it's a trade off, but imo comfort is more important if you've got someone around who can deal with a flat (generally rare unless you are running bald tyres or don't spot the glass).
You can often get them second hand from dealers with some basic warranty, but it can be much cheaper to buy private, generally these days they're fairly reliable and the most common things to need replacing are the tyres and batteries both of which can be picked up far cheaper online than from the dealers (IIRC we were buying batteries for my mum's scooter at ~80 each via a store that specialised in scooter and "recreational" batteries for the likes of boats vs 150ish each, and the tyres were nearly half the cost).
My mother for most of her last 25 years went with Shoprider Sovereigns from memory as the basic design didn't change much (although the controller got better/smaller) and parts were relatively easy to get, it was also at the time she bought her last one available from about £1k new, so £500 second hand would definitely be doable, the big issues is that it was a larger unit than the one wozzer has posted.
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If you do get one with soft/inflatable tyres it's worth keeping a set of tools for changing the tyre/inner tube in the back of the car
With my mother's scooter I kept a couple of full spare wheels (the advantage of keep buying the same model) at home, along with a brick and and the necessary spanners. I could do the whole "AA" thing in about 5 minutes when she got a puncture once I got to where she was (hoik the scooter onto the brick, undo the centre nut, swap the wheel and do it up again).
*Just remember to replace them before the grip pattern wears down...(same goes for inflatable ones, and if you remember to replace them before they start to go bald they don't get anything like as many punctures).