Q: Can't delete music from my iPod shuffle

Have you tried following?

If the iTunes sidebar (along left side of window) is not visible, from the menu bar, under View, select Show Sidebar.

Select the shuffle in the sidebar, under DEVICES.  Click on the small triangle to the left of the shuffle name, to drop down its contact list (indented below the shuffle).  Click on Music there.  The list of songs on the shuffle appears to the right.  Select the song you want to remove from the shuffle, and press Delete on the keyboard.  You can select more than one song at the same time.

Are you using the Manually manage music setting (on the iPod's Summary settings screen)?  Instead, IF you are using the Sync Music setting (automatic syncing), you cannot manually add/remove songs on the shuffle directly (from the sidebar).  You need to go to the iPod's Music settings screen in iTunes to change how iTunes automatically syncs songs to the iPod.

Also, IF the shuffle is currently set up to Sync Music with a different iTunes library (not your current iTunes library), you can't change its content unless you associate it with your current iTunes library.  Associating it with a new iTunes library erases its existing content (from the old library) and replaces it with content from the new library.

In the latest iTunes, the iPod's settings screen is accessed by clicking its device button.  Here's a screenshot of my iTunes window (upper left corner) showing an iPod's device button (circled in red)

On the iPod's settings screen, along the left side (in the sidebar), under Settings, click Music.  The iPod's Music settings screen is shown to the right.