I can help with places that are horribly dangerous, but it's a matter of opinion as to what is the most dangerous. Earlier this year I was part of a mission sent by the JCWI to assist refugees. We went to Calais and to Paris. Armed escorts accompanied us every step of the way.
Go to the Stalingrad Metro stop (Northeastern Paris). When you come out turn right and head Northeast to the Avenue de Flandre. Proceed up the Avenue a few blocks and turn left on to any side street. There you will find a somewhat post-apocalyptic venue where migrants have created a camp site. It's bloody dangerous and borderline foolhardy to stroll around without an escort.
And if that's not adventurous enough, there is an abandoned railway running around the centre of Paris called the 'Petite Ceinture' (i.e., 'little belt'). Portions of the track path have been claimed for ad-hoc, makeshift migrant camps. These are essentially lawless.
To try it, you can pick up the tracks near the Gare d'Orsay (or really anywhere along the track path). Walk along the track path towards the Southeastern part of Paris. You'll find tents and mattresses and what-not strewn about and you'll be well satisfied that you have experienced urban danger in Paris. Don't even think about it at night.
(The Guardian, fair use)
All of this is time sensitive because the citizens are really angry and the city is taking steps to fix things. So go now. And as for police statistics? LOL! These places are 'unregulated'.
If you're just looking for medium spooky scary, try the Catacombs with an unofficial night tour guide.
Jonathan Reez asks "Is it okay to visit that migrant camp if not carrying any valuables?"
Unless you have a sign in about 12 languages that you are not carrying valuables, I wouldn't suggest it. Your best strategy is to wait until you spot a police patrol on foot and ask if you can join them. Or stay close behind them without permission, but then expect a sharp challenge. Are you a trafficker? A pimp? Dealing drugs? Looking for children to kidnap? A serial killer? What are you doing there? You don't belong! These questions would by posed to YOU, the visitor, rather than to the inhabitants. The police would be upset because you are in an unregulated place and they will have to waste time escorting you to safety.
Sumydra comments: "-1 for stereotype (migrants -> crime) without any data to back it up"
Interestingly, there has been no mention of 'migrant crime' at all in this answer, even through all of its various edits. The answer addresses 'danger'. Let's turn to an earlier definition put forth on this site where the topic was Russia...
Having said that, if your definition of 'dangerous' includes feeling
unsafe, or being physically intimidated, or any other source of
discomfort brought on by the locals, then these things could lead to a
feeling of being threatened. At the very least the discomfort brought
on by a perceived threat will spoil your mood, and possibly lead to
the conclusion that it's dangerous. This can happen without any crime
being committed, such as a direct physical assault. Physical
intimidation is also not a crime, but leads to the same feeling of
threat and being unsafe.
Source: Safety - Visiting Russia...
This passage is saying that 'danger' is often a state where the person feels the absence of safety. "Crime" and "criminality" can, and often have nothing to do with it. Also, there is nothing in this answer that stereotypes a particular ethnic group. These notions were introduced by commenters, most likely as a misconceived display of piety. It's my experience that the areas mentioned in this answer represent hundreds of groups, including born and bred Frenchmen.
And for those who think such things matter or perceive an insult or otherwise think I am throwing shade, note that I am an Islamic طالب علم
under the tutelage of a Sunni scholar.
Media links helping to define 'dangerous'
Most credible reports inform us that these incidents were sparked by yobbos out on a wilding rather than the migrants themselves.
Update 29 Oct 2016
Related to the closure of the migrant camps in Calais, some news sources are reporting that the Paris migrant camps have expanded to include the 10th Arrondissement. The 10th was not very safe to begin with because of the Gare du Nord and Gare de l' Est. Accordingly, heightened situational awareness is advisable whilst in the 10th (in addition to the previously mentioned 19th).