Something I've been seeing quite a bit of confusion about on forums, and had me puzzled for some time, too:
How do you perform a music search with Cortana in Windows 10 Mobile?
To do a music search with Cortana in Windows 10 Mobile you must either press the microphone button inside Cortana or else hold the Search button so that she enters listening mode. Then a music icon will pop up in the top right corner of the screen. Tapping on this will start a music search. Note that unless Cortana is in listening mode the icon will not show up. You can also ask Cortana "What song is playing?" to start a music search.
EDIT: Cortana has been updated in the latest build of Windows 10 Mobile to always have the music icon in the top right corner.