Typically "on special" or "on sale" is used in Australia.
Specials are usually referred to groceries/perishables and change weekly.
Help your self to some of that ham mate, I got it on special.
Sales are usually refer to periodic shopping events for consumer goods.
Myer are having a sale on men's shirts, you should check it out.
Got the new telly on sale at JB HiFi for only 600 bucks.
Signs outside of grocery shops might say "Weekly specials". In the other example it might say "Winter Sale" or "Clearance Sale".
Discount is generally used to refer to cheaper pricing due to lesser quality (think like a dollar store in USA). The items could be old stock or nearing expiration. "Discount clothing" generally has minor faults, like poor stitching, less commonly it could just refer to last season clothing, but most boutique stores would avoid using the word "discount" for fear of being associated with "lesser quality".
It's also common for single items in grocery stores to be marked as "discounted" if they're slightly damaged or nearing expiration. This is different to being on special as it might only be one or two single cartons of milk that are marked with a discount sticker as opposed to all of a particular type of milk being "on special" for the week.