Did you know there are two kinds of cops in Paris? The gendarmes, who are part of the army, and the Paris police, who are unionized. The police are national, not local, and most of the cops you see in Paris are not from Paris. If you ask them for directions, they'll puzzle over the map along with you.
In the U.S., the cops are often overweight but tough-looking. They ride around in cars all day, they face criminals with guns, and they've seen a lot. In Paris, they somehow look younger and more naïve. They're usually skinnier too.
on horseback in the pedestrian zones
on foot, especially in touristy areas or when there is any kind of demonstration of crowd. They are pretty mellow and usually just watch; I've never seen them intervene, even when people are throwing cherry bombs or being rowdy.
Embassy duty. I always wonder if this is what happens when you displease your superior.
The CRS, or riot police. They are armored and have plexiglass shields. You see them a lot whenever there is a big demonstration in Paris (pretty much every week). When the teachers marched on the education ministry, I counted thirty of their dark blue vans along the boulevard Saint-Germain. There hasn't been any major trouble in Paris since the 2005 suburban riots though.
Yesterday there was a national day of homage to two young policemen killed on the Périphérique (Paris ring road). A drunk driver with no license and multiple convictions was being chased by another police car, and theirs put itself in his path to slow him down. Instead, he ran into them at high speed and killed them instantly. Twelve policemen have died in the line of duty in Paris in the past 20 years. It's a lot, but not compared to any large American city.