Which is the correct way to say it?

You can say both of the American ones, I know you can say I am Japanese, but "I'm a Japanese" sounds wrong - can't expalin why, though!

"I'm a Japanese" is 100% correct.

Mexicans: "I'm a Mexican."

Americans: "I'm an American."

Chinese: "I'm a Chinese."

Japanese: "I'm a Japanese."

I'm a Japanese sounds like the word is an adjective without its noun, when I read it I am left thinking A Japanese ............ what?

so that is why I still dont think it is 100% correct!

Suzi, you are entirely 100% wrong. "Japanese" is also a noun when referring to a person, the same with any other nationality.

You don't say "He's a Japanese person," you say "He's a Japanese." You wouldn't say "He's a Mexican person," would you?

It's the same with the word "Mexican."

Japanese food/Mexican food

He's a Japanese/He's a Mexican

They

1) I'm Japanese
2) I'm a Japanese
3) I'm a Japanese person
Since 'Japanese' can be a noun or an adjective, all three are correct. I suspect the usage of the word (and other similar adjectives; Australian, American etc.) as a noun has come about due to the ommission of 'person' because it, the word 'person', is assumed.
Sorry, but I can't help adding that to my ear 'I'm Japanes

I cant be 100% wrong in a perception, "I'm a Japanese" does sound wrong to me, even if the grammar books say it is right!
and Mik

I guess I was just so astounded that it sounded "wrong" to you I went overboard in my accusation that you were incorrect. Maybe you just never knew any Asian people while growing up? I certainly don't tell foreigners that "I'm an American person." I say "I'm an American." Naturally, the Japanese will say, "I'm a Japanese" instead of "I'm a Japanese person" when introducing themselves.

it must be something about the --- ----ese ending that makes it sound more like an adjective, compared to the -----------an ending

I'm British. I don't say "I'm a Briton", I say "I'm British". If I was Japanese, I think I would say "I'm Japanese" not "I'm a Japanese". The latter is not incorrect, but nor is it colloquial.

After a little research on this issue I was able to find a couple of opinions that suggested that Americans are a little more relaxed about using adjectives as nouns than the majority of English users.
Of course adjectives used as nouns are not uncommon, especially when referring to categories of people; the poor, the rich, the gifted etc.
Perhaps whether the country name begins with a