Check if a post is in specific category to show an icon

Add post_class() to the post container div:

<div id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
    <!-- post markup here -->
</div>

…which will give you, among other classes:

• .category-{ID}
• .category-{slug}

You can then target those classes via CSS to add category-specific icons.