A small Nissan was my choice for all of them. I found ones through a dealer with low milage. I thought they were exceptionally reliable and I didn't want my daughters to break down late at night. They were a 1000 cc and sometimes I was able to hand them down to the next one as student became wage earner. They scraped and bashed them with gay abandon. The last child to have one had it stolen and we never saw that one again. He had failed to put the crook lock on. We insured them on our own insurance but I think those days are gone now.
DS1 is 19 and passed his test at 17 but I didn't buy him a car as he was going to uni. Most unis are not keen on first years taking a car. He is insured as an additional driver on my car and I will get him a car in the next year or two or when he starts work. DS2 about to start lessons. We are very rural and many parents do buy cars. The models of choice seem to be Corsas, Ford Ka, Fiestas. One boy got a very powerful Audi but wrote it off and nearly a car load of people within a week.
DD was a couple months shy of her 18th so bought her a little 1.0 12v Corsa (second hand) which was her 18th birthday present. The insurance was £1800 (more than the car cost us) but my insurance company quoted me that to add her as a named driver on my car so at least she was earning no claims.
She would have rather had a brand new Fiat 500 but that wasn't going to happen. It served its purpose and now at 19 she has part exchanged it for a 1.2 car.
I would recommend checking the insurance before committing to buy a car. Looked at a 1.0 car and was quoted £2,500 for the insurance. It took me a few times to realize that the make and engine size made such a difference at her age. I was looking at it as if it was me buying where the type of car isn't going to make a great deal of difference to my insurance.