I know from here here that I have to say Master of .... and master's degree. But let's consider another situation:
- I was half the way through my master, or
- I was half the way through my master's.
Another example is:
- My first year of Master covers a wide variety of subjects, or
- My first year of Master's covers a wide variety of subjects
The first question is about the 's and the second is about the capitalization of the first letter in Master.