If you want to display or want to change your default product listing with random listing, just change following simple code.
The product listing page has already category filter applied, check the code at app/code/core/Mage/Catalog/Block/Product/List.php, you simply need to add the random filter there.
It is best practice to override your core file in your local directory instead of changing in core file. So copy file
from app/code/core/Mage/Catalog/Block/Product/List.php to app/code/local/Mage/Catalog/Block/Product/List.php
Now open this file Search _getProductCollection() function. In this function find out
$this->_productCollection = $layer->getProductCollection(); and comment this line and add following code just after this line...
$collection = $layer->getProductCollection();
$collection->getSelect()->order('rand()');
$this->_productCollection = $collection;
Now clear your cache and you are done.
The product listing page has already category filter applied, check the code at app/code/core/Mage/Catalog/Block/Product/List.php, you simply need to add the random filter there.
It is best practice to override your core file in your local directory instead of changing in core file. So copy file
from app/code/core/Mage/Catalog/Block/Product/List.php to app/code/local/Mage/Catalog/Block/Product/List.php
Now open this file Search _getProductCollection() function. In this function find out
$this->_productCollection = $layer->getProductCollection(); and comment this line and add following code just after this line...
$collection = $layer->getProductCollection();
$collection->getSelect()->order('rand()');
$this->_productCollection = $collection;
Now clear your cache and you are done.