Magento – How to Add Additional Sort By Option

First add a dummy attribute called 'discount_percent' and make it "used for sorting" (just as Giuseppe said in the other answer).
It's not important what type it is or what values it has.
it's easier that way. If you don't want to add the attribute you have to rewrite the method Mage_Catalog_Model_Config::getAttributeUsedForSortByArray to include your attribute. I would rather avoid rewrites as much as possible. Now you need to tell magento to take a special action when the sorting is done by that attribute.
For this you have to rewrite (I don't know other way) the method Mage_Catalog_Model_Resource_Product_Collection::addAttributeToSort()
You will need a new module for that. let's call it Easylife_Sorting.
Here are the files.

app/etc/module/Easylife_Sorting.xml - the declaration file

<?xml version="1.0"?>
<config>
    <modules>
        <Easylife_Sorting>
            <codePool>local</codePool>
            <active>true</active>
            <depends>
                <Mage_Catalog />
            </depends>
        </Easylife_Sorting>
    </modules>
</config>

app/code/local/Easylife/Sorting/etc/config.xml - the configuration file

<?xml version="1.0"?>
<config>
    <modules>
        <Easylife_Sorting>
            <version>1.0.0</version>
        </Easylife_Sorting>
    </modules>
    <global>
        <models>
            <catalog_resource>
                <rewrite>
                    <product_collection>Easylife_Sorting_Model_Resource_Product_Collection</product_collection>
                </rewrite>
            </catalog_resource>
        </models>
    </global>
</config>

app/code/local/Easylife/Sorting/Model/Resource/Product/Collection.php - your class override. Here is where the magic happens.

class Easylife_Sorting_Model_Resource_Product_Collection
    extends Mage_Catalog_Model_Resource_Product_Collection {
    public function addAttributeToSort($attribute, $dir = self::SORT_ORDER_ASC){
        //don't screw up the admin sorting
        if (Mage::app()->getStore()->getId() == Mage_Core_Model_App::ADMIN_STORE_ID){
            return parent::addAttributeToSort($attribute, $dir);
        }
        if ($attribute == 'discount_percent') {
            //determine the discount percent value
            $this->addExpressionAttributeToSelect('discount_percent_value', '(({{price}} - final_price) / {{price}})', array('price'));
            //sort by the discount percent value
            $sortDir = $dir;
            //in case you need to reverse the order so when you sort ascending you will see the most discounts first do this:
            //if not then remove the if-else statement below.
            if ($sortDir == self::SORT_ORDER_ASC) {
                $sortDir = self::SORT_ORDER_DESC;
            }
            else {
                $sortDir = self::SORT_ORDER_ASC;
            }
            $this->getSelect()->order(array('discount_percent_value '.$sortDir));
            return $this;
        }
        //otherwise do nothing special
        return parent::addAttributeToSort($attribute, $dir);
    }
}

This works for sure for simple/virtual/downloadable products. I have no idea what effect has on configurable/bundle and grouped products.

for -> Recently Viewed see here

for ->Sorting by rating

Copy the file

app/code/core/Mage/Catalog/Block/Product/List.php to

app/code/local/Mage/Catalog/Block/Product/List.php

in list.php find for this line

$this->_productCollection =$layer->getProductCollection();

which will be in around line no 86 add the following code after that

$this->_productCollection->joinField('rating_summary', 'review_entity_summary', 'rating_summary', 'entity_pk_value=entity_id', array('entity_type'=>1, 'store_id'=> Mage::app()->getStore()->getId()), 'left')

now copy

app/code/core/Mage/Catalog/Model/Config.php to

app/code/local/Mage/Catalog/Model/Config.php

in config.php find for this code

$options = array(
    'position'  => Mage::helper('catalog')->__('Position')
);

replace with

$options = array(
    'position'  => Mage::helper('catalog')->__('Position'),
    'rating_summary' => Mage::helper('catalog')->__('Rating')
);

-->>for BESTSELLER

follow this procedure create a folder naming Inchoo and inside that folder place Catalog and inside catalog create 3 folders Block,etc and Model In Block add Product in Product add List and in List create a file and name it as Toolbar.php and ad this code into it

<?php
class Inchoo_Catalog_Block_Product_List_Toolbar extends Mage_Catalog_Block_Product_List_Toolbar
{
    public function setCollection($collection)
    {
        parent::setCollection($collection);

        if ($this->getCurrentOrder()) {
            if($this->getCurrentOrder() == 'qty_ordered') {
                $this->getCollection()->getSelect()
                     ->joinLeft(
                            array('sfoi' => $collection->getResource()->getTable('sales/order_item')),
                             'e.entity_id = sfoi.product_id',
                             array('qty_ordered' => 'SUM(sfoi.qty_ordered)')
                         )
                     ->group('e.entity_id')
                     ->order('qty_ordered ' . $this->getCurrentDirection());
            } else {
                $this->getCollection()
                     ->setOrder($this->getCurrentOrder(), $this->getCurrentDirection())->getSelect();
            }
        }

        return $this;
    }
}

now in etc folder create a file with name config.xml and add this code

<config>
    <modules>
        <Inchoo_Catalog>
            <version>0.1.0</version>
        </Inchoo_Catalog>
    </modules>
    <global>
        <blocks>
            <catalog>
                <rewrite>
                    <product_list_toolbar>Inchoo_Catalog_Block_Product_List_Toolbar</product_list_toolbar>
                </rewrite>
            </catalog>
        </blocks>
        <models>
            <catalog>
                <rewrite>
                    <config>Inchoo_Catalog_Model_Config</config>
                </rewrite>
            </catalog>
            <catalog_resource>
                <rewrite>
                    <product_collection>Inchoo_Catalog_Model_Resource_Product_Collection</product_collection>
                </rewrite>
            </catalog_resource>
        </models>
    </global>
</config>

Now in Model create a file naming Config.php and add this code.

<?php class Inchoo_Catalog_Model_Config extends Mage_Catalog_Model_Config
{
    public function getAttributeUsedForSortByArray()
    {
        return array_merge(
            parent::getAttributeUsedForSortByArray(),
            array('qty_ordered' => Mage::helper('catalog')->__('Sold quantity'))
        );
    }
}

also create Resource folder in Model and in Resource folder create Product folder and create a file naming Collection.php and add following code.

<?php
class Inchoo_Catalog_Model_Resource_Product_Collection extends Mage_Catalog_Model_Resource_Product_Collection
{
    protected function _getSelectCountSql($select = null, $resetLeftJoins = true)
    {
       $this->_renderFilters();
       $countSelect = (is_null($select)) ?
           $this->_getClearSelect() :
           $this->_buildClearSelect($select);

       if(count($countSelect->getPart(Zend_Db_Select::GROUP)) > 0) {
           $countSelect->reset(Zend_Db_Select::GROUP);
       }

       $countSelect->columns('COUNT(DISTINCT e.entity_id)');
       if ($resetLeftJoins) {
           $countSelect->resetJoinLeft();
       }
       return $countSelect;
    }
}

Now finally activate this module by going to app/etc/modules create a file Inchoo_Catalog.xml add this code.

<?xml version="1.0"?>
<!--
/**
 * Magento
 *
 * NOTICE OF LICENSE
 *
 * This source file is subject to the Academic Free License (AFL 3.0)
 * that is bundled with this package in the file LICENSE_AFL.txt.
 * It is also available through the world-wide-web at this URL:
 * http://opensource.org/licenses/afl-3.0.php
 * If you did not receive a copy of the license and are unable to
 * obtain it through the world-wide-web, please send an email
 * to license@magentocommerce.com so we can send you a copy immediately.
 *
 * DISCLAIMER
 *
 * Do not edit or add to this file if you wish to upgrade Magento to newer
 * versions in the future. If you wish to customize Magento for your
 * needs please refer to http://www.magentocommerce.com for more information.
 *
 * @category    Mage
 * @package     Mage_Connect
 * @copyright   Copyright (c) 2014 Magento Inc. (http://www.magentocommerce.com)
 * @license     http://opensource.org/licenses/afl-3.0.php  Academic Free License (AFL 3.0)
 */
-->
<config>
    <modules>
        <Inchoo_Catalog>
            <active>true</active>
            <codePool>community</codePool>
            <depends />
        </Inchoo_Catalog>
    </modules>
</config>

and for SALE I suggest you this extension as I cannot find any programmatic way to achieve this.