For my situation, I have several animations attached to a model. One for walking, one for running. Each part of the model has its own set: [leftfoot]Walk, [leftfoot]Run, etc...
The quickest way to remove them all was to go to the NLA Editor and they are all listed there to be easily deleted.
This first requires you to use the solution of @user14463 to clear the current model's animations for all parts.
Next, open the NLA Editor and all the animations are listed there. Select each one, and press X to delete it.
However, I found that this does not always work.... for some reason I just cannot get rid of these actions, and they seem to multiply when I do things like copy objects around, and they end up with phantom objects referring to them.
So, an easy way to get rid of an action is to open the python console.
For example, I had the following actions:
I don't know where the .001, .002, and .003 versions came from, but I don't want them!
In the console:
# Get the collection of actions
actions = bpy.data.actions
# Print all the action names
for actionname in actions.keys():
# (printout omitted)
# And now, remove action "[StomperChassis]Walk.001"