WP_Query to get posts in a specific category and post format

Two problems i see:
change aside to post-format-aside
and since you are using foreach loop change new WP_Query( $singargs ); to get_posts( $singargs ); so your code sould look like this:

<?php 
$singargs = array(
'numberposts' => 1,
'tax_query' => array(
    'relation' => 'AND',
    array(
        'taxonomy' => 'category',
        'field' => 'slug',
        'terms' => array ( 'gift-of-the-day' ),
    ),
    array(
        'taxonomy', => 'post_format',
        'field' => 'slug',
        'terms' => array( 'post-format-aside'),
    )
)
);
$singPost = get_posts( $singargs );
foreach ( $singPost as $post ) : setup_postdata($post); ?>
<aside>
    <h2><?php the_title(); ?></h2>
</aside>
<?php endforeach; wp_reset_postdata(); ?>