What term describes a printed list or public display that shows options for diners to select?

The correct term for a printed list or public display that shows options for diners to select is "menu." A menu is a crucial component of any restaurant, particularly in quick-service settings, as it serves not only to inform customers about the available food and drink options but also plays a key role in the overall dining experience. It typically includes descriptions of each dish, which helps customers make informed choices.

In contrast, a billboard is primarily an advertising medium used to promote businesses or events, which does not specifically cater to dining options. A catalog often refers to a systematic, organized list of items, usually relating more to sales and inventory than to dining. A guide, while it may provide information about various options, does not specifically imply a selection of food or drink like a menu does. Therefore, "menu" best captures the essence of what diners interact with when selecting their meals in a restaurant.