How can I see which coupon codes customers are actually using?

There is a Coupon-wise Report that shows, for each coupon code, the number of Uses, the Discount given and the Revenue. It opens on the last 30 days; change the dates with the date picker and use Export CSV to download it.

Its old Reports menu is hidden from the sidebar at the moment, so type Coupon Wise in the admin search to open it.

For order-by-order detail, go to Analytics → Reports and open Discounts by order (in Finances). It lists each sale with its gross sales, discounts and net sales, and you can add the Discount code column to see which code was used.