iPhone 5s calendar not saving events
I had this problem. I drove 75 miles to nearest Apple store with an assigned appointment time just to have the "tech" flip a setting button from "Show" to "Hide".

In other words if you press "Hide" it hides the calendar, but in pressing "Hide" it changes the button to "Show". So when I was looking at the button which says "show", I assumed it was in show mode. Silly me. You have to press show to make it actually show, but then the button will display as "Hide", as if you're in hide mode, but you're not. It's an action button. Press for Show or Press for Hide. Whatever is displayed, your phone is doing opposite to that.

The button can be found by pressing Calendars to bring up a list of your calendars, then pressing the relevant Show button for the calendar. Alternatively, you can press the Show All Calendars button to unhide all the calendars at once.