I don't know much about France, but in Denmark people definitely notice some groups' criminality more than others. There are basically two main groups that contribute a disproportionate amount of Denmark's crime: 1) poor immigrants, and 2) poor Danes. The media seems to like railing against #1 a lot more than #2, though. It's no longer politically incorrect to stereotype immigrants in general, but it's still taboo to stereotype poor, uneducated Danes; you'll be accused of being just an elite Copenhagener who looks down on "white trash".

Much of Denmark's violence (though there isn't a lot of violence) comes precisely where these two groups meet each other and get into fights, over for example controlling the drug trade. Shootouts between biker gangs (who are almost exclusively Danish) and immigrant gangs (who are obviously not) are some of the few times you will find an actual shooting in Denmark. If gangs of Danes are fighting gangs of immigrants over control of an illegal business, I personally blame both groups, not only the immigrants.