Aside from writing "Saint Patrick's Day" or "St Patrick's Day", which is truly the correct way to spell the shortened form? Is it 'St Paddy's Day" or "St Patty's Day"?
The Irish sometimes call it "St Paddy's Day" (or sometimes just "Paddy's Day"). I have certainly never heard anyone refer to it using "Patty". That's not a contraction of Patrick. It's a contraction of Patricia, and therefore a woman's name.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.