Hi,
You can add such icons with the Products>Badges system.
In the "Quantity" setting, you can define the quantity you want the badge to appear for and all the products with that quantity will get the badge. So if you put 0 in there, all the products without stock will get the badge.
But you can say "items in stock" in that. So you need to do it in a diverted way:
- create an unpublished category that will be added as second category to the products without stock
- add it to the "available" badge
- create a mass action via the menu System>Mass action with a trigger like "every hour", and a filter on the product_quantity field being different from 0 and an action to add that category to the products.
- create a mass action via the menu System>Mass action with a trigger like "every hour", and a filter on the product_quantity field being equal to 0 and an action to remove that category to the products.