Hitch-hiking ones nobody believes, from the sixties and early seventies.
A friend's first:
He was hitching to Manchester from London, standing at Hendon, when he was offered a lift by someone he vaguely recognised but couldn't put a name on. The bloke was going about two stops up the road, but better than nothing so he got in. They started to chat and got on really well, and when they got to the turnoff the bloke said "Look, where are you going?" and on being told "Manchester" said, "I fancy a drive. I'll just have to ring my wife and tell her what I'm up to," found a phone at a petrol station and presumably did that after fuelling the car, then drove him to Manchester, talking all the way and having a wonderful time. It was Spike Milligan.
And some of mine (there are lots, hitch-hiking used to be like that, infinite possibilities):
I was walking towards the A64 in York carrying a Dansette gramophone in one hand and a bag in the other so I wasn't yet hitching, when a (really nice!) car pulled up beside me and the man in it asked if I wanted a lift. He was going out along the A64 so I said yes, and got in. When we got to the motorway turning he said he'd have to let me off there because he was going south (I hadn't told him where I was going, you didn't, that way you could say "here's where I get off" if you needed to), so I said, "But so am I!" and we went on down the motorway together. He was going to London, it turned out, and so was I. When we got to Scratchwood services and stopped for coffee he asked where I'd like to be let off, and I said, "Oh, anywhere near the West End, I can get a tube from there," but he said he'd take me to my home if I liked, and when we were nearing Camden he asked where to go, and turned out to live on the same street as me but he kept the car in a garage so I'd never seen it. I had a terrible time persuading him not to tell that story as a good joke all over the street: I wasn't meant to be in York that weekend, I'd been collecting the record-player illicitly and my parents didn't like me hitching.
Another time I was going North from Hendon and a coach pulled up beside me. There were several people there waiting for lifts (back then it was just before the motorway and the last safe place to stand with your thumb out, so people took turns) and he beckoned us all up to the door and explained he was driving a lot of tourists around, but they had taken a special train up to Edinburgh and he had to go to meet them there and was going up the road empty. Lifts for all! He stopped at every services up the road to pick up anyone hitching at them, and several turnoffs to drop people, and we had a wonderful time, with more than one guitar being played and so on. He left me at the A64 -- bit of a pattern emerging but I didn't live in York.
There was the time I got a lift wearing a new pair of jeans whose side-seam had gone all down the outside of one leg while I was away from home and had no spare pair with me, so it was held together with safety-pins. I complained about it a bit, and said I was going to take them back and try for a new pair but I didn't suppose I'd get one... He asked which shop and I said, "Millets in Oxford Street, so I don't think I have a snowball's chance when I've been wearing them," and he told me not to worry about it and took me there. He was the managing director of Millets, and there was no trouble about me getting replacement jeans from them!
Then there was the lift from a Scotsman with a set of bagpipes on the back seat, going down to Dartmoor for a fortnight's holiday because his mother wouldn't let him practise them where the neighbours might hear....
I enjoyed hitching. The only dodgy lift I ever had was going south out of London on the A23, a man whose driving was so terrifying that I asked him to let me out at a set of lights, suggesting once I was safely on the the pavement that he really needed to sleep it off. I got another lift quite quickly, and I saw his car in a lay-by a little further along the road with him asleep in it and was very glad he had taken my advice.