Hi from Russia.
How to Add a Post’s Category Name to Body Class?
I need it for customize menu.
I found only this example:

function add_category_name($classes = '') {
   if(is_single()) {
      $category = get_the_category();
      $classes[] = 'category-'.$category[0]->slug; 
   }
   return $classes;
}
add_filter('body_class','add_category_name');

Can this code help me? Where i have to insert this code? functions.php?

Where did the place you found the code snippet suggest you put it?
Did you try it?

Here
I did not try, because I do not know where to insert this code.

That link tells you where to put the snippet.

Thanks!!! It works!!! Now in body classes i can see class - "category-name" and in css i added some style for this category. Cool!!!