Half Man Half Biscuit. Not sure if you would call them funny songs or just stupid songs. This one is about an obscure Eastern European football team who don't exist anymore.
Warren Zevon did some wryly amusing stuff as well. Get onto Spotify and check out "exciteable boy", "Roland the headless thompson gunner", "lawyers guns and money" etc.
Junior Brown sung a song called "My wife thinks you're dead" which describes an inconvenient encounter with an ex...
And Billy Connolly's "talkin' blues (what's in a name)" which features the story of a "lovely couple, the name was McCann, the gave birth to a child, a fine young man..." whom they christen Jenghis. "Now life's pretty hard as an ordinary man, but it must be pretty hellish as Jenghis McCann!"