To remove an action you need to disable Fake user option with F button and then click X button. After that you will see no active action in action editor (That will have only one button to create new action + New). Clicking on button that opens list of actions you will see action that you removed with 0 at the beginning of this name (like 0 MyUselessAction). It means that has 0 references on it. Saving your scene and loading it again will remove that dangling reference from .blend file
You have two options. One is to go to the NLA Editor and remove the action by pressing the delete button on the selected action (you might need to click the drop down button on the parent animation). The second option is to go to the Hiercy(objects panel) and select your object with the animation, click the plus icon to the left of the animations sub menu and right click on the action you want to remove, then you can see a delete option, follow by clicking it and it should be removed. If you find your action only listed in the NLA Sheets menu, then you can only use the first option.
In the Actions Editor, Shift + Left Mouse click on the 'x' button (next to the shield icon or even next to it), all the actions-data will be cleared from the file.
Go to Outliner menu (the one on the top right corner in animations tab) and click on the display mode button (left to the search bar), it will show a drop - down menu and click on Blender File option, now you can see actions. Right-Click on the specific action which you want to delete and press delete. It's permanently gone!