[noun] + なしで [noun] + なしに [noun] + なし [noun] + なしの + [noun] [noun] + なしでは / なしには + [negative]
なし means "none" or "without", and it goes straight after a noun. With で it says how you do something – without that thing; なしに is the stiffer "without"; on its own at the end of a phrase it is "no ~".
なし follows a noun, never a verb. For "without doing", a verb takes its ない-form and で, or ずに.
With a noun the two often swap. なしで is the everyday form and suits a tool or a resource you managed without. なしに sounds more formal and is more at home in writing; it suits what should have come first – permission, notice, a word to the people affected – and reads as a complaint that it didn't.
Bare なし, at the end of a phrase or before です, is the short "no ~" you see on signs, forms and menus. Its opposite is あり: ちゅうしゃじょう駐車場あり is "parking available", ちゅうしゃじょう駐車場なし "no parking".
Followed by a negative, the pattern says the thing is required: "can't … without". なしには is the written form, なしでは the spoken one.
These go beyond なし after a single noun.
Before another noun, なし takes の, the way a noun does.
A verb can reach なしに only through こと, in the dictionary form: ことなしに. That is a literary N1 pattern with its own entry; in speech, ないで or ずに does the job.