I was never a big fan but I can claim to have been to her most unusual live performance.
In late 1983 (November, I think) she was on a European tour with a small band and was booked to play at the Royal Albert Hall. That was probably somewhat ambitious and ticket sales were so poor that the gig was cancelled. Unfortunately, no one thought to tell Melanie, who arrived in London all set to perform.
So, rather than sit in her hotel sulking, she put word out through the London folk clubs - one of which I was running at the time - that she would perform outside the Albert Hall instead. As I lived not too far away, I went along. About 100 other people turned up, gathering outside the main door. Melanie turned up, along with a couple of guitarists, and announced that she was going to play in front of the Albert Memorial, in the park opposite the hall. That's in Kensington Gardens, which was closed for the night. So we all climbed over the fence and she began to play. Before she had even finished the first song, a bunch of very aggressive police arrived and ordered us back out of the park, so we all climbed over the fence again.
Someone mentioned that there was an open space behind the Hall, so we went there and, for about 90 minutes, she and her two colleagues played the set that they would have played had the gig gone ahead. She obviously wasn't familiar with The Wurzels' version of Brand New Key and seemed rather bemused as we all sang the 'Oooh-Ars' but seemed to be thoroughly enjoying herself and, after winding up with 'What Have They Done to My Song', stayed around for another hour or so chatting to fans and signing autographs.
It was a lovely thing to have experienced but, apart from a short report and photo (with me visible) in the following day's Daily Fail, it went largely un-noticed.