Which is the correct way of writing trying to write Serial Number with a pound sign instead of the word? "Serial#" or "Serial #"
Neither usage is standard. The number sign is used as as abbreviation for "number" only when it prefixes a number. This Wikipedia article gives two examples of appropriate uses:
a #2 pencil (read: "a number 2 pencil")
symphony #5 (read: "symphony number 5")
I would be confused by either "serial #" or "serial#". If you must abbreviate, I much prefer "serial no."