How to set up URL alias patterns? I would like to set the URL patterns in such a way that the page name of the parent page gets inserted into the URL. For example, in the Main Menu I've got an "About Us" link (about-us) and I would like to create a child page Contact Us (contact-us) underneath it.  

How do I set up the URL alias patterns so that the child page URL is automatically set to about-us/contact-us?
URL aliases is supported in Drupal by help of pathauto module; if you enable the Token module, you can use it efficiently by using tokens. Once these modules are enabled, you can visit admin/config/search/path/patterns to set the patterns for URLs.

But as I see in your question, you are looking to set this for menu; you can't do that here. 

There is logic behind it for not supporting it. Suppose you have a node/18 for about-us and you have two menu 'About-Us" &amp; "About-Company". Now you can add this same node/18 to both places as child.

How can you have two URL aliases for single node/18? So this is the reason why menu are not supported I think.

You can go for creating content type &amp; use pattern for that. You can set pattern for taxonomy, users, etc.