"stay home" vs. "stay at home"

It varies. Some sentences you can skip "at", and other sentences you can't. For example:

(be at home/at work) I'll be at work until 7:00, but I'll be at home all evening.

(be/stay home - you don't use at) You can vist me anytime. I'll be home all evening.

Both are correct. There are instances where they mean the same thing and some instances where they don't.

If you were telling someone that you didn't intend to go out tonight, you could use either.

"Do you want to go with us to a restaurant tonight?"
"No, I think I'll stay home."
"No, I think I'll stay at home."

However, if someone were asking where you were staying, where the answer might be "a hotel" or "a friend's house", you would definitely say "at home".

"Are you going to get a hotel room for the conference?"
"No, I'm going to stay at home."
"No, I'm going to stay home."

(Note that if you did say the struck-through version, that would tend to imply that you weren't going to go to the conference at all.)

However, that "home" is uncommon in that it also functions as an adverb. You can never leave the preposition out with other similar nouns. For example, this is clearly wrong:

"I'll probably stay work."

In that case, you have to create an adverbial prepositional phrase:

"I'll probably stay at work."

The key question here is whether it is permissible to omit the preposition; I wasn't able to find many references to this other than a business writing instructor who encourages the omission of obvious or superfluous prepositions.

I would suggest that is it permissible to leave it out if the meaning is unchanged and there is no ambiguity introduced.

Home doesn’t refer to a building. When you intend to say you will be going to or staying at a building, you should use the word “house” instead

E.g: i’ll go back to the house or I’ll stay back at the house

Home usually shouldn’t be used with at or to although we frequently use it with stay-at-Home (which is now considered correct)

Eg: I’ll stay home, I’m on my way home, I’m driving home, I’ll be home, I’ll go home

https://www.google.com/amp/s/dictionary.cambridge.org/amp/british-grammar/house-or-home

I'm an American, but I'll definitely go along with the British and their use of "stay at home". It just sounds grammatically wanting and uneducated to throw away the preposition. One sounds like a hick, so to speak, using "home" as an adverb when it should always be a noun, never an adverb or an adverbial noun. (A former Latin/English teacher, if you please.)

P. S. The word "home" cannot be compared with the word "work" in that their usage is so dissimilar. Never let a good preposition go to waste! (LOL)

Language has been devloped to be used conveniently especially in American English. If we could make it short,the shorter, the better.

'Home' also has the original meaning of being an adverb, it's easier not to use prepostions.I recommend not to use prepositions such as 'at' or 'to' for home like Elvis Presley song lyrics "I'll be home."