For the flat, wide pasta and the dish made from such pasta, North American English speakers use lasagna. English speakers from outside North America use lasagne. Other than the spelling, there is no difference between the words in English.
The word comes from Italian, of course. In that language, lasagna is the singular noun and lasagne is the plural. 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.
Examples
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]
The lauded Canadian coffee-and-doughnut company launched fast-food lasagna bowls in October, and espresso drinks in December. [CBC]
The original intent was it be on foods that are risky after the date, perhaps a chilled lasagne. [New Zealand Herald]
