Go to AMG (allmusic.com) and enter the song title into the search bar at the top of the screen. Click on the arrow next to "Artist/Group," select "Song" and click the "Go" button. Click on the appropriate song title. The author of the song will be listed in the "Composer" column.
Go to copyright.gov and click on "Search Copyright Records" and "Search the Catalog" if the song you are looking for came out after January 1, 1978. If the song you are looking for came out before then, click on "Search Request Estimate," fill out the form, including the title of the song, and click on the "Submit" button.
Search the databases of performing rights organizations like ASCAP (ascap.com), BMI (bmi.com), or SESAC (sesac.com). These organizations represent the majority of songs by popular music artists.
Open a music program like Windows Media Player or iTunes and enter the title of the song in the search bar of the program if you own the MP3 of the song. Adjust your view options (Apple-J in iTunes) to see composer information. The "Composer" column reveals the author of the song.