Everyone should be aware by now that French law requires you to carry photo ID issued by your government at all times. (edit - NATIONAL government)
This (for most people) means passports. I know that for many people carrying your passport around is worrying. I've never understood why, and never had a problem with carrying it when I was out and about. I have always had photocopies of my passport, but they have been left in my accommodation as proof (if needed) that I do (or did) indeed have a passport.
Yesterday my ID was asked for by the police doing random checks. They actually stopped the bus I was travelling on and asked everyone (even the driver) for ID. My French ID card was scrutinised minutely, the guy at the back of the bus carrying photocopies only must have had an even more worrying time, looking North African and carrying only a photocopy of his ID (couldn't see what it was).