As noted above, the finacial websites like Yahoo Finance or Google Finance have pretty good search capability for finding stock symbols. However, there are a couple of resources that allow you to look up stock symbols by designated stock exchanges, alphabetically or reverse lookup using the symbol.
The NASDAQ website has a symbol lookup that covers NYSE, NASDAQ and over-the-counter stocks. The system also allows searching for mutual fund and option symbols. Here is the link for the NASDAQ Find Stock Symbols webpage.
Another very useful resource for stock symbol information is the eodata.com website. The site's Home Page has links for stock symbol lists for the major U.S. stock exchanges, foreign stock exchanges, futures trading, mutual funds and options.
Both of these stock symbol resources offer the ability to download the selected lists of symbols. For someone doing in-depth research into companies and their stock symbols, the download feature would allow the data to be used in different ways.