The Ubuntu Wiki presents these instructions to set up scrobbling with Rhythmbox:
Setting up Last.fm Syncing
- Select Edit > Plugins
- Select the box for the Last.fm plugin, then click Configure
- Insert your Last.fm username and password
- Click Close on the Last.fm Configure and Plugin windows, then use Rhythmbox as normal. The played song information will be uploaded automatically to your Last.fm account.
However, there is no Configuration button in the Rhythmbox version shipped with Ubuntu 16.04:
Preferences only sets up which services to scrobble to:
Where exactly should I register my Last.fm credentials?
Last.fm should appear in the left side menu under last.fm, click on it and you will be prompted to login in the main window
It's a bit awkward but it's below "Radio" in the Library section.
Full disclaimer this is 3.4 on Arch Linux, not Ubuntu. Disregard this comment if it's different on Ubuntu.
This version of Rhythmbox implements an alternative scrobbling mechanism that requires an authorisation step from Last.fm through the web browser. Unfortunately, this mechanism is failing (possibly because it is not requiring the user credentials), and Rhythmbox is unable to scrobble to Last.fm.
This issue is now reported as bug 770521 and the work on a fix is on going. Until a backport for version 3.3 is released there is no way to scrobble for Last.fm with this version of Rhythmbox.
I second the suggestion for Sayonara by cptmashek for those willing to keep scrobbling the music they listen to Last.fm.