Difference between Mage::getResourceModel() vs Mage::getModel() vs Mage::getSingleton() in Magento

Mage::getSingleton()

Mage::getSingleton() will first check if the same class instance exists or not in the memory. If the instance exists then it will return the same object from the memory. So Mage::getSingleton() is faster than Mage::getModel().

Example

$product1 = Mage::getSingleton('catalog/product');
$product2 = Mage::getSingleton('catalog/product');
$product1 and $product2 both will share same memory of OS and return only one instance each time.

Mage::getModel()

Mage::getModel() will create a new instance of an object each time even such an object exists in the configuration.

Example

$product1 = Mage::getModel('catalog/product');
$product2 = Mage::getModel('catalog/product');
$product1 and $product2 both have different instant of same object and also occupy different memory

Mage::getResourceModel()

As far as I know, all collections in Magento are resource models. They are instantiated by

Mage::getResourceModel()

or

Mage::getModel()->getCollection()

It doesn't really matter which function you use; the latter one simply calls the first one. The Magento team simply chose to make collections part of the resource, probably because collections need to query the database a lot. Usually, you will not have to call Mage::getResourceModel() for anything else than collections.