Is popcorn vegetarian?
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Cooked in a plant based fat (say, olive oil) and with no added butter, then yes. If you added butter any other sort of animal product or cooked it in an animal fat (say, bacon fat), then no.
Popcorn on it's own is vegan, being just corn. Popcorn toppings may or may not be. If you are buying microwave popcorn read the label, most of them are fine (having butter, but not say, bacon bits), but can contain some rather suspicious things (that are gross even if they aren't animal derived). If you are making your own (which is easy) you can decide what you want to add to your corn. If you are buying it at the movie theatre I would recommend against it.
You can make your own popcorn in your microwave by purchasing your popcorn, putting about 1/2 cup in a brown paper lunch bag, adding whatever oil and seasonings you want (I love coconut oil) and popping!
About butter -- yes for vegetarians, no for vegans.
About popping corn in a paper bag -- I have read warnings against it because of a fire risk. We have a designated microwave popcorn popper (Rubbermaid Microwave Cookables, if they still make it) which makes great naked popcorn. I have had fires in the microwave with other things, where intense heat concentration and flammables are together too long.