How can I find out what items a user has purchased?

Query the table wp_wpsc_purchase_logs, it contains all information about previous purchases.

add this to your themes functions.php:

/*********************************************************
Get purchased articles by user_id or for the current user
*********************************************************/
function haet_recently_bought_articles($user_id=0){
    if($user_id==0)
        $user_id=get_current_user_id();

    //only if there is an active user, otherwise we would get all purchased items from unregistered users
    if($user_id>0) {
        global $wpdb;
        $sql = $wpdb->prepare("
                SELECT prodid, name, price
                FROM `".$wpdb->prefix."wpsc_cart_contents` 
                INNER JOIN `".$wpdb->prefix."wpsc_purchase_logs` ON purchaseid = ".$wpdb->prefix."wpsc_purchase_logs.id
                WHERE  user_ID = %d
                ORDER BY date DESC"
                ,$user_id);
        $items = $wpdb->get_results($sql,ARRAY_A);
        return $items;
    }
    return null;
}

and add the following lines to the template to show the products e.g. page-.php

<ul class="articles">
    <?php 
    $articles = haet_recently_bought_articles();
    foreach($articles AS $article){
        echo '<li>'.$article['name'].'</li>';
    }
    ?>
</ul>

global $wpsc_cart;  

This is the global variable , looping through this will get all the info ie products added in cart by user, just do

 print_r( $wpsc_cart; )

and you will come to know.