Display Child Categories of Current Post ID

For actual categories:

<ul> 
<?php
    global $post;
    // grab categories of current post
    $categories = get_the_category($post->ID);
    // define arguments of following listing function
    $args = array (
        'child_of' => $categories[0], // current post's (first) category 
        'title_li' => '' // disable display of outer list item
    );
    // list child categories
    wp_list_categories($args);
?>
</ul>

The same as above with a custom taxonomy:

<ul> 
<?php
    global $post;
    // grab terms of current post, replace taxonomy name
    $terms = get_the_terms($post->ID, 'name_of_custom_taxonomy');
    // define arguments of following listing function
    $args = array (
        'child_of' => $terms[0], // current post's (first) category 
        'title_li' => '', // disable display of outer list item 
        'taxonomy' => 'name_of_custom_taxonomy' // replace as well
    );
    // list child categories
    wp_list_categories($args);
?>
</ul>

References:

• get_the_category()
• get_the_terms()
• wp_list_categories()