Shoosh / shush is surely an example of a word that's much more often uttered than written. Someone who has heard the word throughout childhood, with the vowel sound of good might assume it's spelled shoosh. Though if it's written shush it could still be pronounced the same, rhyming with push and bush.
Anyhow, as Ozzyjack reports some dictionaries do allow for the shoosh spelling. Another dictionary that allows that as a variant is the Canadian Oxford dictionary, though other dictionaries from the Oxford stable don't.
The shoosh variant is more likely to appear in Australia than elsewhere, though shush is also used here. And shoosh is occasionally used in British and US publications.
(An expression which I think is unique to Australia is a bit of shoosh, meaning some peace and quiet, as in this comment from Gizmodo Australia: "If you travel, commute or just like to get a bit of shoosh from time-to-time then these might be the headphones for you.")
The word can be an interjection, noun or verb. "One celebration that had Bomber fans out of their chairs was when he shooshed the crowd, raising his finger to his mouth." So I'll add shoosh, shooshes, shooshing and shooshed as rare words.