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Is it:
"Neither John nor I has read that book." OR
"Neither John nor I have read that book."?

Thanks.
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Good question. I like "has," but I hear both. (In other words, I don't know the answer.)
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Supposedly, when a compound subject is joined by "or" or "nor", and each of the subjects takes a different verb form, the verb should agree with the subject closest to the verb. (I'm not quite sure how often people actually follow that rule, though.)

There is a good write-up about subject-verb agreement here:
http://grammar.ccc.commnet.edu/GRAMMAR/sv_agr.htm
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Comments  
AnonymousIs it:
    "Neither John nor I has read that book." OR
    "Neither John nor I have read that book."?

Thanks.

I was taught to use 'have'. 
 Yankee's reply was promoted to an answer.
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Thanks, Yankee, now I remember hearing that rule!