You can do this via the desktop application:
1. Hover your mouse over the three dots next to the artist's name, album, or track title.
2. Select 'Share' from the drop-down menu.
3. From the second menu, select 'Copy Spotify URI.'
Spotify recently hidden "Copy Spotify URI" menu item. You probably have the latest app if you don't see this option on your share menu.
To show 'Copy Spotify URI', When hovering over the share menu, hold down the option key on a Mac or the alt key on a Windows computer to receive your URI.