What kind of things do Oxbridge music admissions tutors look for in a candidate?

Basically, at undergraduate level, the most important thing is to be critical of received knowledge. It's this knack for critical thinking that tutors are really looking for. For example, this could mean challenging the way music history is written: is it really possible to see music history as a logical progression from (for example) Bach to Schoenberg? Who writes these histories and what are their biases? Whom do these histories exclude (women, non-Europeans, non-whites etc)? Why is music history always told as the story of composers and not of performers or listeners? This is just an example, but the point is to be critical of the received wisdom that is so often taken for granted. It’s this kind of thinking that Nicholas Cook's Music: A Very Short Introduction does such a good job of introducing.

The most important thing about your personal statement is that it shows you are the kind of person who is able to think in this way. Of course list your interests and activities etc, but it’s important when saying what you’re interested in academically that you show that you’re able to think about those things critically. At interview, you'll have the chance to demonstrate this abilities on the spot. Demonstrating your ability to think in this way is more important than having all the answers.