I’m trying to show the same thing on the checkout page. I’ve tried the code Mike suggested above in the cart-shipping.php file, but it doesn’t work.
I’ve got 4 shipping classes set under the flat_rate shipping method and I’d like to show the shipping class name next to each one on the checkout when the user buys products with each shipping class. Right now it just shows ‘Flat rate’ for each item, but it would be better if it also shows the class name.