The RSPCA (UK-based, Royal Society for the Prevention of Cruelty to Animals) has a publication called Complete Dog Care Manual, which includes a section headlined
Cleaning up dog mess.
So I'd go for that if you're not sure how others might react to various different words (for example, poop may be considered "childish", shit may be a bit too crude, and waste is somewhat "clinical").
Logically, you might think dogs' mess would be more "correct", but in fact that form is quite uncommon.
Note that whenever possible, corporate/government documents, newspapers, etc., tend to avoid directly referring to the excrement itself. You probably wouldn't want to bring it up in after-dinner conversation anyway, but "There's a problem with dog fouling in our local park" might be more acceptable there.
UPDATE: Intrigued by some of the comments, I dug deeper. It turns out Americans favour "poop"...
(NGrams, "American English" corpus)
...whereas Brits favour "mess" or "dirt"...
(NGrams, "British English" corpus)
Note that I didn't include "shit" in those charts. It's actually far more common than all the alternatives put together, but in many contexts it would definitely be far too vulgar to use safely.