Below, we have listed several sites with music or sound recordings that you may be able to use in your projects without getting permission from the copyright holder. Be sure to check the copyright/license status and be sure it matches with your use - it is your responsibility to determine that your use of 3rd party materials is legal!
Wikimedia Commons is a media file repository making available public domain and freely-licensed media content. Wikimedia Commons has a page where you can browse all of their Creative Commons licensed audio files, including music, spoken word, and other sounds.
This library contains recordings ranging from alternative news programming, to Grateful Dead concerts, to Old Time Radio shows, to book and poetry readings, to original music uploaded by users. Many are available for free download.
Musopen is a 501(c)(3) non-profit focused on increasing access to music by creating free resources and educational materials. They provide recordings, sheet music, and textbooks to the public for free, without copyright restrictions.
SoundBible.com offers free sound clips for download in either wav or mp3 format. The Royalty Free Sounds can indeed be used for commercial uses. The free sounds can only if they are under the Creative Commons Attribution, or Public Domain License.
Freesound is a collaborative database of sounds released under Creative Commons licenses.
BBC Sound Effects Library (on CD)
This collection of 40 CDs has a wide variety of sound effects that can be easily ripped for use in a digital project. The sounds are licensed to be used in audio, video, and film productions. To find sound effects, use this list and note the CD and track numbers. Ask for the CDs at the Library's Circulation Desk (call number CD 3696)
The Extreme Music Library (on CD)
This collection of 24 CDs has a wide variety of instrumental music that can be easily ripped for use in a digital project. The sounds are licensed to be used in not-for-profit, educational audio, video, and film productions. The catalog records in HoyaSearch include a note with the genre for each CD. Use those notes to find the CD you are interested in and ask for the CD at the Library's Circulation Desk (call number CD 1275 and CD1276).