Remove top links one by one in Magento 1 using local.xml file

 <default>
 <reference name="root">
 <reference name="top.links">
 
 <!-- Removes 'My Account' link -->
 <action method="removeLinkByUrl"><url helper="customer/getAccountUrl"/></action>
 
 <!-- Removes 'Wishlist' link -->
 <!-- for Magento 1.3.x -->
 <action method="removeLinkByUrl"><url helper="wishlist/"/></action>
 
 <!-- for Magento 1.4.x -->
 <remove name="wishlist_link"/>
 
 <!-- Removes 'My Cart' AND 'Checkout' links-->
 <remove name="checkout_cart_link"/>
 
 <!-- To re-add 'My Cart' or 'Checkout' after removing both 
 <block type="checkout/links" name="checkout_cart_link_custom">
 <action method="addCartLink"></action>
 <action method="addCheckoutLink"></action>
 </block>
 -->
 
 </reference>
 </reference>
 </default>
 
 <customer_logged_out>
 <!-- Removes 'Log In' link -->
 <reference name="top.links">
 <action method="removeLinkByUrl"><url helper="customer/getLoginUrl"/></action>
 
 <!-- Remove Register link --> 
 <action method="removeLinkByUrl"><url helper="customer/getRegisterUrl"/></action>
 
 </reference>
 </customer_logged_out>

 <customer_logged_in>
 <!-- Removes 'Log Out' link-->
 <reference name="top.links">
 <action method="removeLinkByUrl"><url helper="customer/getLogoutUrl"/></action>
 </reference>
 </customer_logged_in>