custom post type problem

You can create specific single.php for your custom post types, in the form single-customposttype.php (it’s the same with archive.php, see Template Hierarchy) for example single-artist-name.php. Then in those templates you can do pretty much whatever you want.

UPDATE

Here’s a very simple example of single-artist-name.php :

<?php get_header(); ?>
<?php if ( have_posts() ) while ( have_posts() ) : the_post(); ?>
    <div id="content">
        <h2>Artist : <?php the_title(); ?></h2>
        <?php the_content(); ?>
    </div>
<?php endwhile; ?>
<?php get_footer(); ?>

With this, if you visit www.mydomain.com/miles-davis/ you’ll see for example :

Artist : Miles Davis

Widely considered one of the most influential musicians of the 20th century etc.

And so on, the same single post template would be applied to any artist-name post.

What sounds strange to me is why would you see all post titles when you visit www.mydomain.com/%artist-name%/, that looks more like a taxonomy archive. Are you sure that artist-name is registered as a custom post type?