Sitecore.Master.Index is use for various search functionalities in
Sitecore.Web.Index is use in
CD environment. Except search what are scenarios to use this index?
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The obvious answer is to use the
Web index for any (site) search related functionalities.
There are also other site search like functionalities. You could think of a news articles or product (,or any other) overview, especially if you're including some kind of filtering mechanism.
Other then that the index could be used for any functionality that require some Sitecore items spread throughout the database or based on large quantities of Sitecore items, for example related items based on tags.
As soon as you are looking for one or more Sitecore items that are not direct parents/children/siblings it could, performance wise, be worth to look at using the index.
For some more information regarding (Fast) Queries and Indexing please read Why we should not use fast queries in Sitecore