Ooh, I've had some choice ones over the years.
My second flat in Edinburgh, next door neighbours were drug dealers, and would have giant screaming rows with each other all the time in the stairwell. Oh, and someone came to our door with what looked like a shotgun wrapped up in a rug looking for the guys next door. Which was rather exciting.
Our flat in London was probably the worst - we had to walk through their flat to get to ours (something the landlord had failed to mention, and it wasn't obvious when we viewed it) but they played Bob Marley all the time. So loudly that it was louder in our flat than I would listen to music. And it was the same album. Over and over again. Every day. They moved out after 4 months, and were replaced by some folk that I suspect were up to fairly dodgy dealings, but they were quiet, and friendly, and I was very happy to have them be our neighbours.
Our last flat in Bath had neighbours upstairs who were really old, and would put on Kenny Rogers "While The Feeling's Good" on repeat, (yes the single song) and then have really really loud elderly sex. One memorable night at 1am all we could hear was "**** me! **** me!" over and over again. I will never, ever, be able to hear that song again without retching. They were really nice though, and it didn't happen very often. I just could never look them in the eye though.
Oh, and in Bath, there was a revolting little troll of a woman who had a little dog that she never managed to keep in the garden, and when it got out would stand at the end of her garden and shout "Gizmo! Gizmo? GIZMO! Gizmo? Gizmo! Gizmo... Gizmo! GIZMO! Gizmo! Gizmo. Gizmo! Gizmo? Gizmo! Gizmo... At 2 in the morning. Or 8am on a Saturday. Until Gizmo came back. I always wished Gizmo would run away from her forever, or bite her face off.
Our new flat's fine in the evening, but the people below play loud dance music (on repeat as well - what the hell is wrong with these people?!) during the day on weekdays. As we won't be there, it's not the end of the world. And it's just as loud as the Marley players, but doesn't travel as much. You've gotta love Scottish tenement buildings. Ours was built in 1890, apparently, and is solid as anything, and so sound doesn't really travel too badly between the flats.