Will mounting a camera on the handlebar instead of the helmet make the picture more shaky? I do a short commute in Paris (about 7 km / 30 minutes each way) and I am seriously considering adding a camera (maybe 2) to my bike to be able to report dangerous behaviour and also generally because it's fun and interesting.

I've seen a lot of great footage on the web but I wonder how they get such a stable picture. I guess that helmet mounted cameras produce an inherently more stable picture but I also heard of stabilizing software.

The thing is I don't wear a helmet and I would like to fix my camera to the handlebar. My bike (Cross Areal CR) has suspension in the front, however I'm not sure it's enough to dampen the road shake, though my route is mostly on smooth asphalt (some potholes, some old paved passages).

Does mounting a camera on the handlebar mean that I must invest in a more expensive stabilized camera or do you find that most cameras work just a well? Are there any tricks to getting a stable picture with such a setup?
Cameras on bikes are a trade-off. I run my camera from the handlebars, but it can be very shaky particularly depending on the road surface: on a newly paved road it can be almost perfect, but on cobbles it's almost useless. 

Certainly you can helmet mount, but then you can look quite silly. (The cuboid Go Pros on top of a helmet just look ridiculous.) There are some advantages though, it's definitely more stable (as @cherouvim correctly points out - your neck and body are great shock absorbers) and it means that you are getting the camera to record (more or less) what you are looking at. 

If you're planning on being able to use the footage to report problems, then it is going to be much more flexible to be able to point the camera dynamically. The lack of a helmet isn't the end of the road though, I use a Contour camera and they've got headband, cap and helmet mounts - so while you'll need some form of headwear, it needn't be a helmet. 

Some of the shake can be mitigated with a fast frame rate and post-production, but it'll be easier if it wasn't there in the first place and the best way for that is not to mount it on the bike.