I plugged in a Logitech A-5572A USB audio adapter and it was detected automatically on boot.
To view audio devices, run aplay -L from the command line. It should look something like this:
# aplay -L null Discard all samples (playback) or generate zero samples (capture) default:CARD=Black TI BeagleBone Black, Default Audio Device sysdefault:CARD=Black TI BeagleBone Black, Default Audio Device default:CARD=Headset Logitech USB Headset, USB Audio Default Audio Device sysdefault:CARD=Headset Logitech USB Headset, USB Audio Default Audio Device
I'm using a small amplified speaker for testing, but you could just as easily use a standard headset plugged into the 1/8 inch jack of the Logitech.
To test the speaker without an audio file, you can run speaker-test -D default:Headset from the command line. Be sure to replace the value after -D with a device from the output of your own aplay results. When running this command, you should hear white noise. Here is my output:
# speaker-test -D default:Headset
If you hear sound, you're doing great. I have an mp3 in my current directory named r2_1.mp3. Use mplayer to play the mp3.
# mplayer -ao alsa:device=default=Headset r2_1.mp3
If you don't hear a sound, but you heard sound with the speaker test, take another look at the format of the -ao value.
I found the information here in this Google Groups thread.