how can I remove broken spigot from garden hose end?
TRY a POTATO !
just cut a piece of potato and &quot;screw&quot; the potato into the female end-- it just might grab enough for you to unscrew the broke male threads --   it is safe on your hands and cheap to try - won't damage anything--  I have used this to get out broken light bulb parts in a socket too! GRANDPA taught me this decades ago!