Harry Potter and the Order of the Phoenix
Mundungus Fletcher always has a slight air of filth around him: when Harry encounters Mundungus back at Number Twelve, Grimmauld Place, Harry sees him pulling out a "grimy" (5.23) black pipe that sends out "Great billowing clouds of greenish smoke" (5.23). In fact, the name "Mundungus" is an old word for tobacco (source). And of course, Mundungus's nickname is "Dung" – a.k.a. poo – so he doesn't seem exactly clean.
Mundungus's lack of personal hygiene only underlines the slightly shady nature of his work: he's constantly just outside the law. But he is also a member of the Order of the Phoenix. As Sirius points out, it's useful to have someone like Mundungus on their side because he "knows all the crooks – well, he would, seeing as he's one himself [...] It pays to have someone like Dung around, he hears things we don't" (5.86). Indeed, Mundungus does overhead something that is useful: he is in the Hog's Head Inn when Hermione organizes the first D.A. meeting, and he passes on the news to the rest of the Order. Thanks to Mundungus, Sirius warns Hermione and Harry that they need to find a less obvious place to hold their meetings, since the Hog's Head Inn attracts a lot of unsavory people (like Mundungus).
So Mundungus is surprisingly helpful. But when we first meet him, he is in disgrace. He has "a business opportunity" (5.36) that draws him away from guard duty at Little Whinging the night that the two dementors attack Harry and Dudley. If Mundungus had been there, perhaps he could have cast the Patronus needed to save the two boys. His disappearing act sends Mrs. Figg on a rage, and Mundungus is highly apologetic when he next meets Harry properly at Number Twelve, Grimmauld Place.