For example, use of shortcut ordinal numbers as bullets for a listing (1st, 2nd, 3rd) is casual but acceptable, if that's your writing style. Personally I prefer just using straight numbers (1, 2, 3). However, except in very casual communication, ordinal numbers used in copy that is NOT list format, just straight text like this, are written out (first, second, etc.).
1st: Open the door.
2nd: Walk through the door.
3rd: Close the door
First, open the door. Second, walk through the door. Third, close the door.
I've written before about another shortcut I see often: the use of "&" in business writing (outside of titles and such). It is no harder to type the three letters "and" than it is to shift-7 to create the &, and it is far more professional.
Greater grammar Nazis than I am are predicting a decline in proper grammar and spelling due to the huge younger population accustomed to tweeting and texting. These shortcuts can be viewed as the destruction of proper English (yes, I tend to this view) or to a new written dialect. I certainly wouldn't want to see a resume that says, "I wud luv 2 work w u bcuz im g8 at it ru hiring?" Key is to understand the use of each in context.