It's both. In England we call it a shopping trolley. The Americanism is shopping cart.

In the United States, we call your first item a "shopping cart." The second is "a shopping basket" or "a hand basket." We do not use the word "trolley" the way the British do. In a restaurant, a wheeled device for showing diners a selection of desserts is a "dessert cart." In an office, a wheeled device to allow a computer to be moved easily from one place to another is a "computer cart." In the United States it is used mostly to describe electrically-powered streetcars that get their power from an overhead wire--and there are very few such systems left, so we hardly use it at all.

Hi Robin, Frankie is correct: in the United States, they're known as "shopping carts." The plastic baskets that are carried by hand and do not have wheels are simply "shopping baskets." I hope this helps. Please let me know if you have any more questions =)

In the US it is called a cart and we call the smaller plastic basket -- a basket.