nope he wasnt born in singapore.
and it doesnt seem as if he was in a prim and proper upbrining as a lad..he appears to have been the rebel against that for much of his teenage life...traveling around alot on adventures.
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Early life
Son of Captain Grant Sparrow.
According to the official Disney website and the game based on the film series, Jack Sparrow was born in British colonial India and was once employed by the East India Trading Company. He helmed the Wicked Wench, an EITC merchant vessel, performing odd jobs for Cutler Beckett. However, when Sparrow refuses to transport slaves and instead frees them in Africa, Beckett sanctions the torching (and sinking) of the Wicked Wench, and literally brands Jack Sparrow a pirate. Jack comes to embrace his outlaw status, becoming a successful pirate with no desire to return to life under the command of others. Later, Jack petitions Davy Jones to raise his ship from the ocean floor. He rechristens her the Black Pearl.
Disney Press began a book series for young readers in June 2006, chronicling Sparrow's adventures as a teenager. Written by Rob Kidd, the first three books follow Sparrow and a young, motley crew in a continuing search for the legendary Sword of Cortés. Along the way, they must battle pirates, ancient curses, sirens, and the power of the sea
http://en.wikipedia.org/wiki/Jack_Sparrow#Early_life