Mules don't normally wear horseshoes? Well, they have hoofs right? Either way, I know for a fact they make a lot of noise on paved surfaces, and while some areas of the dungeon in question is more cave dirt oriented, the lower levels for sure are essentially old (and some new) mines and fortress areas cut out of stone.
Also there was this:
They could make sandals, not booties. Cut a circle for the bottom of the hoof, pierce a number of holes around the perimeter, and lace it up to the mule's legs. I'm not sure it'd like it, and I'd probably have the players' be making animal handling checks
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first of all, there's a number of characters with animal handling/empathy skills, so there's no problem there. They do waste one of the characters by having him/her lead the mule at the back
But *this* idea, MULE SANDALS?!?! I'm about to start cracking up again! How about some MULE BIRKENSTOCKS fer crissakes?! To be serious for a sec here, if I thought something like this was doable in the first place, it would have to be fashioned by someone with horse/mule-tending (or whatever ya call it) as a Profession--pref. in the nearby town. Furthermore, it just seems more realistic, in the setting, a ca. 7th-8th century AD Viking/Anglo-Sxaon milieu, that instead of leather, artisans would probably use felt made from wool.
Last point, someone suggested I was looking for a reason not to do this--this is true to some extent because I found the suggestion hilarious in the first place. Just the name MULE MUFFLERS. Secondly, I wanted to see if other DMs have dealt with this before, and how they did so.