English Gurus!! My Name Is Or My Names Are, Which Is Grammatically Correct?

This has caused serious arguments
between me and friends couple of
times so i decided to bring it to the
house.
English Language Gurus and
Linguists,what are your take on the
above phrases,which is better and
grammatically correct when
introducing oneself.
Hopefully we can all learn a thing or
two from your inputs.