/**
 * get Controller name
 */
$this->getRequest()->getControllerName();

/**
 * get Action name, i.e. the function inside the controller
 */
$this->getRequest()->getActionName();

/**
 * get Router name
 */
$this->getRequest()->getRouteName();

/**
 * get module name
 */
$this->getRequest()->getModuleName();

IN CLASS FILES
$this might not work in class (php) files. In this case, you need to use Mage::app().

Here is the code:

/**
 * get Controller name
 */
Mage::app()->getRequest()->getControllerName();

/**
 * get Action name, i.e. the function inside the controller
 */
Mage::app()->getRequest()->getActionName();

/**
 * get Router name
 */
Mage::app()->getRequest()->getRouteName();


/**
 * get module name
 */
Mage::app()->getRequest()->getModuleName();
The above functions (getControllerName, getActionName, getRouteName, getModuleName) are present in the class Mage_Core_Model_Url.

You can explore all requests with print_r.

echo "<pre>";
    print_r(Mage::app()->getRequest());
echo "</pre>";

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