Can a US notary sign Single affidavit?

I live in the US and my fiancee is in Japan but to marry in Japan I need to have proof that I am single to get married. Affidavit of Competency to Marry or a Single Affidavit is needed to do so. Can I have a notary in the US sign it or does it have to be notarized by a US Embassy in Japan?

I don't recall the correct legal term off the top of my head, but "sign" isn't the right one. An attestation I think - checked, just notarization.
I believe you can got your affidavit notarized in the US for use in a different country (namely Japan). I have in the past (for a legal case) had documents notarized in one country for use in another. Plus, if you were doing this in Japan, your visit to the Embassy puts you on US soil anyway.

It's best you ask your fiance to ask the ward office where you plan to register your marriage what will they accept.

Back when I did I was specifically asked that it was done at the embassy, but this was back when the the certificate of no impediment was recently abolished and an affidavit with a notary at the embassy in its place was a new process. So they may be a bit more relaxed about it now but best to check as ultimately they are going to be processing it for you.
Also note I am a uk citizen so the process may differ slightly.

A quick search on the USA embassy in japans website states under japan law the affidavit needs to be done at the USA embassy/consulate in japan.

"Japanese law requires all foreigners who marry in Japan to first prepare a sworn Affidavit of Competency to Marry, affirming they are legally free to marry, from their own country�fs embassy or consulate in Japan."

If you are unsure then contact the USA embassy in japan too to be sure.
I had to time my appointment at the embassy for when I was going to be in japan for my marriage.

what is the legal position of notary in USA ?
if he/she can represent the US government, it may work. but, I think he/she just represent your statement and he/she does not represent the government.

the Japanese government trusts the documents officially issued by the US government. Japan does not care that you tell a lie to the US government. that is your matter between you and your country.
for example, you already are married in USA and you get a Single Affidavit from US government (i.e. from US embassy in Japan). that is a fault of US government, because they are cheated by you.