I know that this is an old thread but anyway I'm quite surprised that everybody recommends Montmartre(Sacre Coeur)at night.
Maybe in the meantime some things have changed.
I just returned from Paris (April 2012) and I had the bad inspiration to go at Montmartre after midnight.
Immediately after leaving the funicular (on top of the hill) I was in the middle of very strange and dangerous looking crowd (looked like thieves and drug dealers to me) so after making a few steps I've decided that it's safer to leave.
A guy with all his teeth rotten (he said that his name was Francois) decided that he is against my idea of leaving and blocked my way and started to look very insistent at my camera bag, then asked for a cigarette.I've replied that I don't smoke and went back to the funicular.
Nothing bad happened but I didn't stayed more that a minute there and still I was close of being robbed.(at least that was my strong impression and usually I'm not wrong)
Therefore, for anybody still reading this thread,I strongly recommend not to go at Sacre Coeur during the night.