Just tested the code below and it work in v1.7 when added to template/catalog/product/view.phtml.
However Mage::registry('current_category');
is only available if when coming to a product page from a category page (not tested, but may also dependent on if you have seo url that contain the category names within the url)
<?php
$_helper = $this->helper('catalog/output');
$_category_detail=Mage::registry('current_category');
echo $_category_detail->getName(); //gives current category name
echo $_category_detail->getId(); //gives current category id
?>
See http://vinayp.com.np/how-to-get-category-name-and-id-on-product-page-on-magento/
To display all the categories that a product belong to do
<?php $categories = $_product->getCategoryIds(); ?>
<?php foreach($categories as $k => $_category_id): ?>
<?php $_category = Mage::getModel('catalog/category')->load($_category_id) ?>
<a href="<?php echo $_category->getUrl() ?>"><?php echo $_category->getName() ?></a>
<?php endforeach; ?>
See http://www.magentocommerce.com/boards/viewthread/27720/