ANYBODY or ANYONE?

Which is correct?:
Is anybody at home? / Is anyone at home?

They mean exactly the same thing.  The only difference is that "anyone" is on a slightly higher register (на чуть более стилистическом уровне).  "Anybody" sounds more informal.  [The same is true with the pairs someone/somebody; no one/nobody; and everyone/everybody.]

In most circumstances you can say the sentences with or without the preposition "at".  That is, you could also say, "Is anybody home?" and "Is anyone home?"  However, in the case where you are right there at someone's house or apartment, knocking on the door, the variant without the word "at" would seem more natural to me:  "Is anyone/anybody home?!"

ANYBODY or ANYONE?

which is correct? or how do you say correctly? 
Is anybody at home? / Is anyone at home?

both are completely acceptable, personally I would say is there anybody at home? or is anyone at home?

ANYBODY or ANYONE?

How to say correctly:
Is anybody at home? / Is anyone at home?

I would use both, but I especially use "anyone"

http://dictionary.cambridge.org/fr/grammaire/grammaire-britannique/anyone-anybody-or-anything

ANYBODY or ANYONE?

How to say correct:
Is anybody at home? / Is anyone at home?

Both are correct ;)

i think the correct is : is anybody home?.

ANYBODY or ANYONE?

How to say correct:
Is anybody at home? / Is anyone at home?