How do I get the attributes of a short code from a post?

Either way you can do it very easily. First Hook a Function. I used wp_head but you can use anyone you want.

add_action( 'wp_head', 'find_shortcode_attribute' );

function find_shortcode_attribute() {
global $post;

if( has_shortcode( $post->post_content, 'nggallery' ) ) {
    preg_match_all( '/' . get_shortcode_regex() . '/', $post->post_content, $matches, PREG_SET_ORDER );

    echo "<pre>";
    print_r( $matches );
    echo "</pre>";

    // you will get the attribute on $matches[3]
    // do whatever you want here.

}

Your solution has helped me find an answer to my own problem of getting the attribute of a certain shortcode, but I fear there may be an issue with your approach.

You're using preg_match to check the post_content, which will only return 1 match. If you have a post that has multiple shortcodes in it, it will only return the first one, which may not be the one you're looking for.

Instead, you should use preg_match_all, then loop through the regex_matches array to check for the shortcode you need.

Also, you don't need to use str_replace to remove the quotes from the string. Wordpress has a built-in function shortcode_parse_atts which takes that a string of parameters (in your case it would be $regex_matches[3]) as an argument and outputs an array. This would also spare you the use of wp_parse_args function.

I have found a solution. After a bunch of searching around I found some code that determines if a post has a specific short code in it.

That code also has some attempt to parse the parameters to the short code. Which is good because I want the gallery ID. There were some issues with the code as posted so I tweaked it. Here is code that can find a short code in a post and get its parameters:

        <?php //  Look for a NextGEN gallery
        $galleryID;
        $previewIndex = 1; 
        $regex_pattern = get_shortcode_regex();
        preg_match ('/'.$regex_pattern.'/s', $post->post_content, $regex_matches);
        if ($regex_matches[2] == 'nggallery') :
            //  Found a NextGEN gallery find out what ID
            //  Turn the attributes into a URL parm string
            $attribureStr = str_replace (" ", "&", trim ($regex_matches[3]));
            $attribureStr = str_replace ('"', '', $attribureStr);

            //  Parse the attributes
            $defaults = array (
                'preview' => '1',
            );
            $attributes = wp_parse_args ($attribureStr, $defaults);

            if (isset ($attributes["id"])) :
                $galleryID = $attributes["id"];
            endif;
            if (isset($attributes["preview"])) :
                $previewIndex = $attributes["preview"];
            endif;
        endif;
        ?>

What needed to be tweaked was the handling of the parameters. Using trim instead of some secret charter and switching to wp_parse_args to correctly handle the short code parameters. Once the above code finishes executing in a WP loop $galleryID will hold the NextGEN gallery ID and previewIndex will be set to the preview index or 1 if no previewIndex attribute was present.

previewIndex is an attribute I "added" to indicate what thumbnail to use for the gallery preview. NextGEN ignores it and the galleries render as normal but now I can use it for my Theme to display a specific icon in the preview entry.

Here is the code from my loop-index.php and loop-category.php that handles creating the gallery preview:

            <?php /* Enhance the content preview with an image from the NextGEN gallery */ ?>
            <?php
            global $nggdb;
            $gallery = $nggdb->get_gallery ($galleryID, 'sortorder', 'ASC', true, 0, 0);
            $image = $gallery[$previewIndex];
            $total_images = count ($gallery);
            ?>

            <?php if (isset($image)  &&  isset($image->thumbURL)) : ?>
                <?php /* Show the thumbnail */ ?>
                <a href="<?php the_permalink(); ?>" title="<?php printf( esc_attr__( 'Permalink to %s', 'twentyten' ), the_title_attribute( 'echo=0' ) ); ?>" rel="bookmark">
                    <img class="entry-preview-image" src="<?php echo $image->thumbURL ?>" align="left" />
                </a>
            <?php endif; ?>

            <?php /* Show the text excerpt */ ?>
            <?php the_excerpt(); ?>

            <?php /* Show the statement of number of images contained */ ?>
            <em><?php printf( _n( 'This gallery contains <a %1$s>%2$s photo</a>.', 'This gallery contains <a %1$s>%2$s photos</a>.', $total_images, 'twentyten' ),
                'href="' . get_permalink() . '" title="' . sprintf( esc_attr__( 'Permalink to %s', 'twentyten' ), the_title_attribute( 'echo=0' ) ) . '" rel="bookmark"',
                number_format_i18n( $total_images )); ?>
            </em>
        <?php endif; ?>

This makes use of information from another answer regarding accessing NextGEN gallery objects in order to get the thumbnail and the count of images in the gallery.