Stabling Costs (Rules)
The Tack, Harness, and Drawn Vehicles Table gives the cost of stabling a mount as 5sp per day.
Lifestyle Expenses (Interpretation)
Take a look at the descriptions of the Lifestyles in the Lifestyle Expenses section of the PHB. In particular, look at the jump between Modest and Comfortable (emphases mine).
Modest. A modest lifestyle keeps you out of the slums and ensures that you can maintain your equipment. You live in an older part of town, renting a room in a boarding house, inn, or temple. You don't go hungry or thirsty, and your living conditions are clean, if simple. Ordinary people living modest lifestyles include soldiers with families, laborers, students, priests, hedge wizards, and the like.
Comfortable. Choosing a comfortable lifestyle means that you can afford nicer clothing and can easily maintain your equipment. You live in a small cottage in a middle-class neighborhood or in a private room at a fine inn. You associate with merchants, skilled tradespeople, and military officers.
These descriptions, in my opinion, indicate that choosing a modest lifestyle means you would need to find and pay for stabling for your mounts and beasts of burden separately and in addition to your own accommodations. Choosing a comfortable lifestyle means accommodations for such animals can assumed to be included within your lifestyle. A day laborer would not have a mount, nor would their boarding house have an attached stable. A military officer would likely have a mount, a merchant would have either a mount or beast of burden, and there would be facilities for these at their homes (or places of work, if these were different).
Quasi-medieval verisimilitude (Rules-free)
A low quality inn might not have a stable at all. If it did have a place for mounts, it might just be a fenced yard with no roof, or as Darrel Hoffman points out, simple hitching posts.
A high quality inn likely has a stable with individual stalls.
Inns cater to travelers and often have attached stables. There is unlikely to be a stables independent of such a business, that is, a place whose sole income comes from boarding mounts unrelated to traveling people, simply because the meagre income from such an enterprise would not approach what could be gained from other uses of valuable property in a city, and in a smaller town where land was less valuable there would not be enough demand for the service of stabling mounts (however, see Peter Cordes' comment on livery stables as a place to rent, rather than board, horses, and the possibility of renting extra stalls at such an establishment). Nearby a sports arena or hippodrome, riding course for the wealthy, permanent cavalry encampment, or other place where those wealthy enough to own mounts would want them kept close to their place of use, there might be enough demand to have an independent mount boarding business.
Along trade routes there are likely to be stockyards unattached to inns where you could board beasts of burden like your cargo mule. As Patrick Artner points out, villages and towns might have a commons or village green, where livestock would be allowed to graze for free. In your case, this might not be a particularly safe place to keep your mule, and the use of the common land might be restricted to citizens and forbidden to travelers.