When it comes up, people usually seem to think I must have a full name and will ask if it's
Katherine or
Caitlin (to which I explain, neither).
I really don't see the need to have Katherine/Caitlin as a "formal" name if you're going to use
Kate as your spoken/social name. It is classic and works fine as a stand-alone name (somewhat akin to my perspective of
Emma).
I think it's a little plain for a middle name (which I usually like to spruce up). Maybe it would work well with an interesting first name.
My only
real complaint is that people seem to think
Katie is equivalent to
Kate. I have never, EVER gone by
Katie; when people call me it I explain that
Katie is not my name.
I prefer
Kate to
Cate as I prefer more common spellings to uncommon/unique spellings. However, if one's full name was
Caitlin and uses a "
Kate" nickname, I would assume they spell it
Cait (and the same for similar instances).
Ask any questions!