You can buy stalls by clicking on the ellipsis at the bottom of the screen and choosing the barn icon. After that, you can swipe left or right until you reach an empty stall. Click on the plus sign and you’ll see the option to buy this stall for a small fee.
If you already have a Star Stable Online account, you can unlock one stall by selecting the stall with the Star Stable logo and logging in.
You can also click on the small cash register in the bottom left corner when viewing the paddock, where you can see the option to buy stalls in packs of three or five.
You can have up to eleven different horses at once if you unlock all the stalls!