A new update to Shazam's iPhone app ( no Android) adds the "Auto Shazam" feature introduced to the service's iPad app recently. Flip the switch on and the app will continuously listen to the world around you, grabbing tags for music and TV out of the air automatically. Watch out for your battery, though.
The Shazam update has caused my iPhone 5 to slow down to a near halt. Restarting, uninstalling/reinstalling the app to see if it makes any difference. Otherwise, I'll permanently delete and make do with Soundhound.
Edite: Restarting the phone, and reinstalling the app seems to have done the trick. Haven't had to do that with an app in a very long time. Terrible job, Shazam.