One of my favorite approaches is to create dynamic classes for the body element.
So:
<?php
$app = JFactory::getApplication();
$menu = $app->getMenu();
$active = $menu->getActive();
$class = $active->alias . " pageid-" . $active->id;
?>
<body class="<?php echo $class; ?>">
Some examples of what the above code will produce is:
Home page:
<body class="home pageid-13">
About us page:
<body class="about-us pageid-15">
Now you can create custom specific styles per page, using the above classes.
Update 2023
Getting the menu in Joomla 4:
use Joomla\CMS\Factory;
$app = Factory::getApplication();
$menu = $app->getMenu();
$activeMenuitem = $menu->getActive();
https://docs.joomla.org/Menu_and_Menuitems_API_Guide