I took Rosie's post to exclude clubs, as such, and to be dealing with casual, unstudied exposure to traditional music. In a sense the question might be: is there still an "oral" tradition where one could, if he or she wished, pick up a tune or a song without going to a specialist venue.

Well, when I was a kid, it was music at my paternal grandparents' house - often around their piano on a Saturday night. My maternal grandfather, I have to say, was a well-loved but wicked old man. When I was about 5 or 6, he got me to learn a song to sing in front of the family in the early evening - it was my mother's birthday. So, to the tune of "The Sailor's Hornpipe", in front of all the family, I innocently sang:

Tiddleywinks, old man - get a woman if you can.
If you can't get a woman, then you'll have to get a man.
When the Rock of Gibraltar takes a flying leap at Malta,
You will never catch your bollocks in a corn beef can."

Mum was not amused, but I believe there was a twitch in my father's face - Granddad, of course, roared with laughter. I recalled the song again last year and sang it once more to my Mum - this time she was in the 7th year of being in a care home with dementia, not speaking or moving. So I sang her the song - and saw a definite twitch in her face...

The song, I believe, was a serviceman's song from the WW1 - my grandfather was in the RAMC (Royal Army Service Corps - jocularly known as Rob All My Comrades).

So - you can hear songs passed on by word of mouth in the weirdest circumstances! :-)