You can view your owned items by tapping on the Person/Face icon in the lower right-hand corner in Design Home. Then tap on the Inventory tab, and scroll down to see all of the Inventory items you own.

Also, when designing Live Challenges in Design Home, you can use any item that you own.

To find your owned items, click on the bubbles or requirements in the Challenge. Then, you will be sent to the Owned Section, and you will see the items in your Inventory.