How to get child categories of a given Post

To get the child categories of a given parent category slug, use a combination of get_category_by_slug() and get_categories().

The latter function will return an array of category objects matching the specified query arguments array; the former will return the ID of a category, given its slug.

So, for example:

<?php
$motorbike_child_cat_args = array(
    'child_of' => get_category_by_slug( 'motorbikes' )
);

$motorbike_child_cats = get_categories( $motorbike_child_cat_args );
?>

Then, you can do whatever you want with your array-of-category-objects. For example, to get an array of child-category names:

<?php
$motorbike_child_cat_names = array();
foreach ( $motorbike_child_cats as $child_cat ) {
    $motorbike_child_cat_names[] = $child_cat->name;
}
?>

Really, what you do with it is up to you at that point.

EDIT

If you need to get child categories of an arbitrary post, then you can use get_the_category().

If you’re inside the Loop, simply call get_the_category(); if you’re outside the Loop, then you need to pass the Post ID to the call: get_the_category( $id ).

So, for example, to build an array of names of child categories (regardless of parent) of the current post:

<?php
$my_post_categories = get_the_category();

$my_post_child_cats = array();
foreach ( $my_post_categories as $post_cat ) {
    if ( 0 != $post_cat->category_parent ) {
        $my_post_child_cats[] = $post_cat->cat_name;
    }
}
?>

Or, for example, to build an array of names of ‘motorbike’ child categories of the current post:

<?php
$my_post_categories = get_the_category();

$motorbikes_child_cats = array();
foreach ( $my_post_categories as $post_cat ) {
    if ( 'motorbikes' == $post_cat->category_parent ) {
        $motorbikes_child_cats[] = $post_cat->cat_name;
    }
}
?>

Is that more what you’re looking for?

EDIT 2

If you just need to get all the categories of your post:

<?php
$all_post_categories = get_the_category();

$my_post_cats = array();
foreach ( $my_post_categories as $post_cat ) {
    $my_post_cats[] = $post_cat->cat_name;
}
?>

That will give you all of the categories for the current Post. I have no idea how motorbikes category slug factors into this question.