I know dog's balls is to mean "cool, great". I'm guessing it's a quite recent invention, too. I'd like to ask native speakers: how does the phrase sound to you, what connotations does it bring about? Is it common?
Could you please explain what it means then?
I haven't heard the phrase "dog's balls" in that sense (others may have) but in UK Eng "dog's bollocks" (the same part of the canine anatomy) means "just the best": "This new car of mine is the dog's bollocks". It's like a coarser version of the genteel and dated "bee's knees" or "cat's pyjamas".
The phrase "sticks out like dog's balls" means something is very obvious and plain to see, especially if it looks out of place or inappropriate: "In that street of old wooden houses that modern glass one sticks out like dog's balls".
These are fairly vulgar terms and you would have to be careful where you used them, in some company they would be considered rude, but they can also be effective at the right time and place.
not a teacher
Did not you mean "dog's bollocks" by any chance?
I've never heard of "dog's balls".
(note: I am not a native speaker of English).
how do you even come up with something like bee's knees?
Yes, it is a nice phrase and although I did say that it's a little dated I still often hear it said.
The bee's knees: The phrase was first recorded in the late 18th century, when it was used to mean 'something very small and insignificant'. Its current meaning dates from the 1920s, at which time a whole collection of American slang expressions were coined with the meaning 'an outstanding person or thing'. Examples included the flea's eyebrows, the canary's tusks, and one that still survives - the cat's whiskers. The switch in meaning for the bee's knees probably emerged because it was so similar in structure and pattern to these other phrases.
I agree with JMurray, seems to be a mix or confusion between "the dogs bollocks" which does mean "cool or great", and "Something or someone sticks out like dogs balls", which as is mentioned, means something or someone that is very different from everything else, or is very obvious.
So while Robby Williams thinks he's the dogs bollocks, Lady gaga sticks out like dogs balls where ever she goes.
By the way, this site's just the cats pyjamas.