Remove the membrane from the back of the ribs, if present, by loosening it with a dull knife and pulling it off with a paper towel.
Mix brown sugar, paprika, black pepper, salt, chili powder, garlic powder, onion powder, and cayenne pepper in a bowl to create the rub.
Generously apply the rub to all sides of the ribs.
Place the trivet in the Instant Pot, add water, apple cider vinegar, and liquid smoke, then arrange the ribs standing on their side, wrapped around the pot's interior.
Seal the Instant Pot and cook on high pressure: 23 minutes for baby back ribs or 35 minutes for spare ribs. Allow for natural pressure release.
Broil the cooked ribs with barbecue sauce applied on top until the sauce caramelizes, about 2 minutes.