For the flat, wide pasta and the dish made from such pasta, North American English speakers use lasagna. English speakers from outside North America usually use lasagne.
The word comes from Italian, of course. In that language, lasagna is the singular noun and lasagne is the plural, but this does not carry over into the words’ treatment in English. Both the plural and the singular forms are usually treated as mass nouns, taking singular verbs.
The word first appeared in English in the 19th century, but the dish did not become popular in English-speaking countries (the U.S. first, then elsewhere) until the second half of the 20th century. Both forms have had nearly identical trajectories, in terms of when they appeared and when they grew more common, on opposite sides of the Atlantic.
Finish with chicken with peppers and sausage or a lasagna with farmers market vegetables, ricotta and tomatoes. [Los Angeles Times]
One of my favourite dishes is Mum’s cabbage lasagne. [Guardian]
This classic lasagna is the kind that defines lasagna in the USA: ruffle-edged noodles, creamy ricotta and mozzarella filling, and a rich tomato sauce. [The New Lasagna Cookbook, Maria Brucino Sanchez]
The original intent was it be on foods that are risky after the date, perhaps a chilled lasagne. [New Zealand Herald]
The lauded Canadian coffee-and-doughnut company launched fast-food lasagna bowls in October, and espresso drinks in December. [CBC]
The lasagne is well worth ploughing through. [Lonely Planet Scotland]