Commons' search engine generally works the same as the search engine on Wikipedia. For help with using the search engine, you can visit the English Wikipedia search help page (which is particularly detailed), or the same help page on other Wikipedias.
Search options on the Commons
intitle: and prefix: and other search options are activated on Commons search. 
- Find pages whose title contains fine line. (intitle:"fine line")
- Find pages in file space whose title starts with the three letters c o w (prefix:file:cow)
- A search for 'shark' videos (filetype:video)
- A search for 'flower' where the files are 16 bit (filebits:16)
- A search for 'stars' where the files are PDFs (filemime:pdf)
- Find articles that are not in Category:Musicals but are in Category:1920 (-incategory:"musicals" incategory:"1920")
- Find pages in Help namespace, whose title, text, or media content contains the phrase "panorama". (help: "panorama") You can search in a namespace non searched by default by prepending your keyword with the name of the namespace (function provided by mw:Extension:SearchExtraNS since 2013-02-25).
Working search enhancements on the Commons
- Long image and file names in categories can be activated in preferences in the gadget tab: Special:Preferences#mw-prefsection-gadgets
- Intersections of categories can be shown by default gadget Help:FastCCI
- Intersections of categories and subcategories and keywords via Deepcat Gadget ()
- Search suggestions and search results can be set to open in new browser tabs: MediaWiki talk:Search-results-new-tab.js
Proposed search enhancements for the Commons
- Help:Image searching
- mw:User:Rainman/search internals
- Commons:Editor's index to Commons#Search (COM:EIC#Search) - more search-related entries in the Editor's index
- See interlanguage links on the left side pane for other languages
- Search through categories: see Commons:Categories#Tools
- Commons:Bugs#Search - Bugzilla requests.