Most problems with Local Storage happen because people don't understand that this is an asynchronous operation. Writing/reading data in Local Storage takes some time, especially if the amount of data is big. If you write something, you need to wait for "On item set" event before trying to access this data. A quite common mistake is to save some variables to Local Storage on the end of level, then switch to another layout and try to read these values back from Local Storage without any delay.
Also, to minimize the number of keys and events, you can put all your variables into a dictionary and save/load the dictionary. Here is how I usually do this: