Get a list of existing transients

The db query would look like this:

SELECT `option_name` AS `name`, `option_value` AS `value`
FROM  $wpdb->options
WHERE `option_name` LIKE '%transient_%'
ORDER BY `option_name`

To sort the results by their function (site transients, timeouts) use a function like this:

add_action( 'shutdown', function(){

    global $wpdb;
    $sql = "SELECT `option_name` AS `name`, `option_value` AS `value`
            FROM  $wpdb->options
            WHERE `option_name` LIKE '%transient_%'
            ORDER BY `option_name`";

    $results = $wpdb->get_results( $sql );
    $transients = array();

    foreach ( $results as $result )
    {
        if ( 0 === strpos( $result->name, '_site_transient_' ) )
        {
            if ( 0 === strpos( $result->name, '_site_transient_timeout_') )
                $transients['site_transient_timeout'][ $result->name ] = $result->value;
            else
                $transients['site_transient'][ $result->name ] = maybe_unserialize( $result->value );
        }
        else
        {
            if ( 0 === strpos( $result->name, '_transient_timeout_') )
                $transients['transient_timeout'][ $result->name ] = $result->value;
            else
                $transients['transient'][ $result->name ] = maybe_unserialize( $result->value );
        }
    }
    print '<pre>$transients=" . esc_html( var_export( $transients, TRUE ) ) . "</pre>';
});

Now you get an array, separated by the transient functions with unserialized values.

Sample output:

$transients = array (
  'site_transient' => 
  array (
    '_site_transient_browser_0f2bbce5647f9c092edea85f1b5d9145' => 
    array (
      'platform' => 'Windows',
      'name' => 'Opera',
      'version' => '12.02',
      'update_url' => 'http://www.opera.com/',
      'img_src' => 'http://s.wordpress.org/images/browsers/opera.png',
      'img_src_ssl' => 'https://wordpress.org/images/browsers/opera.png',
      'current_version' => '11.64',
      'upgrade' => false,
      'insecure' => false,
    ),
    '_site_transient_browser_4155da8a3756e08080a06133476ef1fd' => 
    array (
      'platform' => 'Windows',
      'name' => 'Firefox',
      'version' => '19.0',
      'update_url' => 'http://www.firefox.com/',
      'img_src' => 'http://s.wordpress.org/images/browsers/firefox.png',
      'img_src_ssl' => 'https://wordpress.org/images/browsers/firefox.png',
      'current_version' => '16',
      'upgrade' => false,
      'insecure' => false,
    ),
  ),
  'site_transient_timeout' => 
  array (
    '_site_transient_timeout_browser_0f2bbce5647f9c092edea85f1b5d9145' => '1352809256',
    '_site_transient_timeout_browser_4155da8a3756e08080a06133476ef1fd' => '1366603648',
  ),
  'transient' => 
  array (
    '_transient_feed_mod_46583134dd8a90321b20eb41cdeb134c' => '1366089834',
    '_transient_feed_mod_57bc725ad6568758915363af670fd8bc' => '1352920456',
    '_transient_plugins_delete_result_1' => '1',
  ),
  'transient_timeout' => 
  array (
    '_transient_timeout_feed_46583134dd8a90321b20eb41cdeb134c' => '1366133033',
    '_transient_timeout_feed_57bc725ad6568758915363af670fd8bc' => '1352963656',
  ),
)